14689:
14249:
14684:{\textstyle {\begin{aligned}1-\Phi \left(x\right)&=\left({\frac {0.39894228040143268}{x+2.92678600515804815}}\right)\left({\frac {x^{2}+8.42742300458043240x+18.38871225773938487}{x^{2}+5.81582518933527391x+8.97280659046817350}}\right)\\&\left({\frac {x^{2}+7.30756258553673541x+18.25323235347346525}{x^{2}+5.70347935898051437x+10.27157061171363079}}\right)\left({\frac {x^{2}+5.66479518878470765x+18.61193318971775795}{x^{2}+5.51862483025707963x+12.72323261907760928}}\right)\\&\left({\frac {x^{2}+4.91396098895240075x+24.14804072812762821}{x^{2}+5.26184239579604207x+16.88639562007936908}}\right)\left({\frac {x^{2}+3.83362947800146179x+11.61511226260603247}{x^{2}+4.92081346632882033x+24.12333774572479110}}\right)e^{-{\frac {x^{2}}{2}}}\end{aligned}}}
6971:
12513:
the total variance of the mean depends on the unknown variance, and the sum of squared deviations that goes into the variance prior (appears to) depend on the unknown mean. In practice, the latter dependence is relatively unimportant: Shifting the actual mean shifts the generated points by an equal amount, and on average the squared deviations will remain the same. This is not the case, however, with the total variance of the mean: As the unknown variance increases, the total variance of the mean will increase proportionately, and we would like to capture this dependence.
14785:
12828:
1481:
6485:
2005:
67:
48:
12525:
hyperparameters, one specifying the sum of squared deviations of the pseudo-observations associated with the prior, and another specifying once again the number of pseudo-observations. Each of the priors has a hyperparameter specifying the number of pseudo-observations, and in each case this controls the relative variance of that prior. These are given as two separate hyperparameters so that the variance (aka the confidence) of the two priors can be controlled separately.
12951:
19843:
14732:
14747:
13041:
19853:
6175:
15277:"It has been customary certainly to regard as an axiom the hypothesis that if any quantity has been determined by several direct observations, made under the same circumstances and with equal care, the arithmetical mean of the observed values affords the most probable value, if not rigorously, yet very nearly at least, so that it is always most safe to adhere to it." â
5349:
5590:
6195:). Many properties of normal distributions generalize to properties of NEF-QVF distributions, NEF distributions, or EF distributions generally. NEF-QVF distributions comprises 6 families, including Poisson, Gamma, binomial, and negative binomial distributions, while many of the common families studied in probability and statistics are NEF or EF.
13052:, can be called the first generator of normal random variables. This machine consists of a vertical board with interleaved rows of pins. Small balls are dropped from the top and then bounce randomly left or right as they hit the pins. The balls are collected into bins at the bottom and settle down into a pattern resembling the Gaussian curve.
5943:
14083:
1318:
15024:
5122:
11111:
13541:
and the normal distribution, since the transform employs just addition and subtraction and by the central limit theorem random numbers from almost any distribution will be transformed into the normal distribution. In this regard a series of
Hadamard transforms can be combined with random permutations
12524:
associated with the prior, and another parameter specifying the number of pseudo-observations. This number serves as a scaling parameter on the variance, making it possible to control the overall variance of the mean relative to the actual variance parameter. The prior for the variance also has two
12512:
To handle the case where both mean and variance are unknown, we could place independent priors over the mean and variance, with fixed estimates of the average mean, total variance, number of data points used to compute the variance prior, and sum of squared deviations. Note however that in reality,
13526:
is faster than the BoxâMuller transform and still exact. In about 97% of all cases it uses only two random numbers, one random integer and one random uniform, one multiplication and an if-test. Only in 3% of the cases, where the combination of those two falls outside the "core of the ziggurat" (a
12930:
in physical experiments are often modeled by a normal distribution. This use of a normal distribution does not imply that one is assuming the measurement errors are normally distributed, rather using the normal distribution produces the most conservative predictions possible given only knowledge
5358:
15641:. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 932.
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11452:
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1454:
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13890:
9691:
15150:
10033:
8481:
1125:
1138:
6170:{\displaystyle \mu \mid x_{1},\ldots ,x_{n}\sim {\mathcal {N}}\left({\frac {{\frac {\sigma ^{2}}{n}}\mu _{0}+\sigma _{0}^{2}{\bar {x}}}{{\frac {\sigma ^{2}}{n}}+\sigma _{0}^{2}}},\left({\frac {n}{\sigma ^{2}}}+{\frac {1}{\sigma _{0}^{2}}}\right)^{-1}\right)}
12508:
Keep in mind that the posterior update values serve as the prior distribution when further data is handled. Thus, we should logically think of our priors in terms of the sufficient statistics just described, with the same semantics kept in mind as much as
12733:
13103:, which is a 12-section eleventh-order polynomial approximation to the normal distribution. This random deviate will have a limited range of (â6, 6). Note that in a true normal distribution, only 0.00034% of all samples will fall outside ±6Ï.
10852:
4620:
14878:
12386:
359:
13585:
13245:
11254:
13411:
5648:
4472:
5344:{\displaystyle D_{\mathrm {KL} }(X_{1}\parallel X_{2})={\frac {(\mu _{1}-\mu _{2})^{2}}{2\sigma _{2}^{2}}}+{\frac {1}{2}}\left({\frac {\sigma _{1}^{2}}{\sigma _{2}^{2}}}-1-\ln {\frac {\sigma _{1}^{2}}{\sigma _{2}^{2}}}\right)}
3296:
9355:
2008:
For the normal distribution, the values less than one standard deviation from the mean account for 68.27% of the set; while two standard deviations from the mean account for 95.45%; and three standard deviations account for
4337:
483:
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12090:
1331:
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558:
5585:{\displaystyle H^{2}(X_{1},X_{2})=1-{\sqrt {\frac {2\sigma _{1}\sigma _{2}}{\sigma _{1}^{2}+\sigma _{2}^{2}}}}\exp \left(-{\frac {1}{4}}{\frac {(\mu _{1}-\mu _{2})^{2}}{\sigma _{1}^{2}+\sigma _{2}^{2}}}\right)}
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10693:
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plotâis a plot of the sorted values from the data set against the expected values of the corresponding quantiles from the standard normal distribution. That is, it is a plot of point of the form (Ί(
4106:
15265:
that was designated for private circulation only. But it was not until the year 1738 that he made his results publicly available. The original pamphlet was reprinted several times, see for example
14093:
with arbitrary precision. The drawback of this algorithm is comparatively slow calculation time (for example it takes over 300 iterations to calculate the function with 16 digits of precision when
842:
14219:
9914:
6560:, in accordance to the central limit theorem. In the bottom-right graph, smoothed profiles of the previous graphs are rescaled, superimposed and compared with a normal distribution (black curve).
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10112:
210:
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1006:
14078:{\displaystyle \Phi (x)={\frac {1}{2}}+\varphi (x)\left(x+{\frac {x^{3}}{3}}+{\frac {x^{5}}{3\cdot 5}}+{\frac {x^{7}}{3\cdot 5\cdot 7}}+{\frac {x^{9}}{3\cdot 5\cdot 7\cdot 9}}+\cdots \right)}
2102:
120:
7847:
1313:{\displaystyle {1 \over 2}\left\{\left({\frac {\sigma _{0}}{\sigma _{1}}}\right)^{2}+{\frac {(\mu _{1}-\mu _{0})^{2}}{\sigma _{1}^{2}}}-1+\ln {\sigma _{1}^{2} \over \sigma _{0}^{2}}\right\}}
993:
12501:
From the analysis of the case with unknown variance but known mean, we see that the update equations involve sufficient statistics over the data consisting of the number of data points and
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computed from the data consisting of the mean of the data points and the total variance of the data points, computed in turn from the known variance divided by the number of data points.
8179:
13517:
The two optional steps allow the evaluation of the logarithm in the last step to be avoided in most cases. These steps can be greatly improved so that the logarithm is rarely evaluated.
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12114:. The test compares the least squares estimate of that slope with the value of the sample variance, and rejects the null hypothesis if these two quantities differ significantly.
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256:
5813:
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themselves, it is necessary to reciprocate, add, and reciprocate the result again to get back into the original units. This is exactly the sort of operation performed by the
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6349:
3812:
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8888:
4504:
15019:{\textstyle f(x)={\frac {2\beta ^{\frac {\alpha }{2}}x^{\alpha -1}\exp(-\beta x^{2}+\gamma x)}{\Psi {\left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}}}
13325:
8118:
7749:
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13099:(0,1) deviates, add them all up, and subtract 6 â the resulting random variable will have approximately standard normal distribution. In truth, the distribution will be
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6267:
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1984:
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is an adjustment constant, which can be anything between 0 and 1. If the null hypothesis is true, the plotted points should approximately lie on a straight line.
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11106:{\displaystyle t={\frac {{\overline {X}}-\mu }{S/{\sqrt {n}}}}={\frac {{\frac {1}{n}}(X_{1}+\cdots +X_{n})-\mu }{\sqrt {{\frac {1}{n(n-1)}}\left}}}\sim t_{n-1}.}
10270:
9151:
9124:
8686:
8659:
7114:
2679:
2627:
1948:
17358:
Halperin, Max; Hartley, Herman O.; Hoel, Paul G. (1965). "Recommended
Standards for Statistical Symbols and Notation. COPSS Committee on Symbols and Notation".
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10774:
9375:
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4821:
3863:
3739:
776:
748:
10847:
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9060:
9034:
2129:
15812:
Geary RC(1936) The distribution of the "Student's ratio for the non-normal samples". Supplement to the
Journal of the Royal Statistical Society 3 (2): 178â184
13534:; i.e., it is equivalent to sampling a real number from the standard normal distribution and rounding this to the nearest representable floating point number.
6884:
6634:
6558:
6230:
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9238:
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8981:
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8508:
8358:
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3719:
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2651:
2599:
2033:
8540:
4875:
is a normal random variable. The consequence of this result is that the normal distribution is the only distribution with a finite number (two) of non-zero
15788:
12997:
the distribution of long duration river discharge or rainfall, e.g. monthly and yearly totals, is often thought to be practically normal according to the
4379:
18501:
1772:
13530:
Integer arithmetic can be used to sample from the standard normal distribution. This method is exact in the sense that it satisfies the conditions of
18086:
Shore, H (1982). "Simple
Approximations for the Inverse Cumulative Function, the Density Function and the Loss Integral of the Normal Distribution".
13863:
function. His algorithms vary in the degree of complexity and the resulting precision, with maximum absolute precision of 24 digits. An algorithm by
13796:{\displaystyle \Phi (x)=1-\varphi (x)\left(b_{1}t+b_{2}t^{2}+b_{3}t^{3}+b_{4}t^{4}+b_{5}t^{5}\right)+\varepsilon (x),\qquad t={\frac {1}{1+b_{0}x}},}
372:
10621:
4247:
16073:
11447:{\displaystyle F={\frac {\left(X_{1}^{2}+X_{2}^{2}+\cdots +X_{n}^{2}\right)/n}{\left(Y_{1}^{2}+Y_{2}^{2}+\cdots +Y_{m}^{2}\right)/m}}\sim F_{n,m}.}
3553:
9904:{\textstyle m_{\alpha }={\frac {\alpha m_{0}\sigma _{1}^{2}+(1-\alpha )m_{1}\sigma _{0}^{2}}{\alpha \sigma _{1}^{2}+(1-\alpha )\sigma _{0}^{2}}}}
1449:{\displaystyle \mu -\sigma {\frac {{\frac {1}{\sqrt {2\pi }}}e^{\frac {-\left(q_{p}\left({\frac {X-\mu }{\sigma }}\right)\right)^{2}}{2}}}{1-p}}}
12148:, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified.
16640:(second revised ed.). Wageningen, The Netherlands: International Institute for Land Reclamation and Improvement (ILRI). pp. 175â224.
5756:{\displaystyle {\mathcal {I}}(\mu ,\sigma ^{2})={\begin{pmatrix}{\frac {1}{\sigma ^{2}}}&0\\0&{\frac {1}{2\sigma ^{4}}}\end{pmatrix}}}
13088:
12003:
3198:
496:
15189:
11851:
10303:
8219:
922:
18630:
17699:
4978:
normal is essential; without it the property does not hold. For non-normal random variables uncorrelatedness does not imply independence.
19856:
19113:
11746:
has a (univariate) normal distribution. The variance structure of such
Gaussian random element can be described in terms of the linear
13877:
after recalling Hart68 solution is not suited for erf, gives a solution for both erf and erfc, with maximal relative error bound, via
7883:
19892:
19021:
16503:
9686:{\textstyle {\frac {1}{\int _{\mathbb {R} ^{n}}X_{0}^{\alpha }(x)X_{1}^{1-\alpha }(x)\,{\text{d}}x}}X_{0}^{\alpha }X_{1}^{1-\alpha }}
2381:
2313:
2245:
11829:â a four-parameter family of probability distributions that extend the normal law to include different skewness and kurtosis values.
10161:
19808:
17882:(1860). "V. Illustrations of the dynamical theory of gases. â Part I: On the motions and collisions of perfectly elastic spheres".
16233:
9696:
15359:"Calculating CVaR and bPOE for common probability distributions with application to portfolio optimization and density estimation"
15145:{\textstyle \Psi (\alpha ,z)={}_{1}\Psi _{1}\left({\begin{matrix}\left(\alpha ,{\frac {1}{2}}\right)\\(1,0)\end{matrix}};z\right)}
4148:
19674:
18886:
18645:
18494:
9440:
3824:
13296:
is a modification of the BoxâMuller method which does not require computation of the sine and cosine functions. In this method,
19887:
19569:
19333:
12978:, and T-scores. Additionally, some behavioral statistical procedures assume that scores are normally distributed; for example,
11836:, also known as the exponential power distribution, allows for distribution tails with thicker or thinner asymptotic behaviors.
1765:
10394:
10028:{\textstyle \sigma _{\alpha }^{2}={\frac {\sigma _{0}^{2}\sigma _{1}^{2}}{\alpha \sigma _{1}^{2}+(1-\alpha )\sigma _{0}^{2}}}}
7981:
19007:
18438:
18327:
18301:
18282:
17939:
17812:
17751:
Lexis, Wilhelm (1878). "Sur la durée normale de la vie humaine et sur la théorie de la stabilité des rapports statistiques".
17688:
17620:
17601:
17508:
17489:
17462:
17443:
17396:
17348:
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17230:
17141:
17122:
17103:
17084:
16645:
16448:
16357:
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16024:
15705:
15646:
17:
16947:
4042:
19328:
19272:
19170:
18932:
18570:
18060:
17425:
15464:
8476:{\displaystyle \ln p(x)=-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}-\ln \left(\sigma {\sqrt {2\pi }}\right).}
3116:
789:
14107:
calculates values of the standard normal cumulative distribution function using Hart's algorithms and approximations with
12938:, results can be made to have a normal distribution by either selecting the number and difficulty of questions (as in the
1120:{\displaystyle {\mathcal {I}}(\mu ,\sigma ^{2})={\begin{pmatrix}1/\sigma ^{2}&0\\0&1/(2\sigma ^{4})\end{pmatrix}}}
19614:
19348:
19201:
18876:
18620:
15575:
15231:
14175:
12909:
12181:
11531:
8213:
170:
19078:
15358:
13546:
12818:
distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.
10052:
19846:
19518:
19494:
19073:
18487:
18337:
17670:
16988:
12576:
Regression problems â the normal distribution being found after systematic effects have been modeled sufficiently well.
5055:
4988:
18391:
15294:
curve saves us from proportioning the merit of discovery between the two great astronomer mathematicians." quote from
12942:) or transforming the raw test scores into output scores by fitting them to the normal distribution. For example, the
5863:
79:
19882:
19715:
19592:
19553:
19525:
19499:
19417:
19343:
18766:
18514:
16613:
16095:
15827:
15738:
14840:
13248:
13119:
12946:'s traditional range of 200â800 is based on a normal distribution with a mean of 500 and a standard deviation of 100.
12935:
12529:
12097:
8689:
4931:
2039:
1758:
1746:
1705:
929:
7785:
3476:
855:
19703:
19669:
19535:
19530:
19375:
19183:
18881:
18635:
12152:
11806:
11560:
11233:
are independent standard normal random variables, then the ratio of their normalized sums of squares will have the
8483:
Since this is a scaled and shifted square of a standard normal variable, it is distributed as a scaled and shifted
4923:
3394:
365:
13527:
kind of rejection sampling using logarithms), do exponentials and more uniform random numbers have to be employed.
13069:) will have the standard normal distribution. The drawback of this method is that it relies on calculation of the
7556:
19453:
19366:
19338:
19247:
19196:
19068:
18851:
18816:
11833:
11641:
8893:
4982:
2135:
1636:
1572:
1131:
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18846:
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18680:
18675:
17276:
15169:
13058:
12786:
In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where
12176:, where in the basic model the data is assumed to be normally distributed, and normal priors are placed on the
11768:
11470:
10463:
10308:
6396:
1684:
1545:
13285:) in these equations; and the angle is distributed uniformly around the circle, chosen by the random variable
12728:{\textstyle {\frac {\partial }{\partial t}}f(x,t)={\frac {1}{2}}{\frac {\partial ^{2}}{\partial x^{2}}}f(x,t)}
11855:
10167:
19897:
19783:
19649:
19357:
19206:
19138:
19123:
19016:
18990:
18922:
18761:
18655:
18650:
18592:
18577:
18467:
17416:
13859:
lists some dozens of approximations â by means of rational functions, with or without exponentials â for the
13018:
12920:. The use of the assumption of normal distribution occurring in financial models has also been criticized by
12494:
From the analysis of the case with unknown mean but known variance, we see that the update equations involve
8543:
6610:
3416:
1833:
690:
12092:. For normally distributed data this plot should lie on a 45° line between (0, 0) and (1, 1).
3654:
19877:
19619:
19609:
19300:
19226:
18927:
18786:
15198:
14807:â the long-standing problem of testing whether two normal samples with different variances have same means;
13081:
article. Wichura gives a fast algorithm for computing this function to 16 decimal places, which is used by
12900:
of exchange rates, price indices, and stock market indices are assumed normal (these variables behave like
12815:
12570:
11177:
11118:
10700:
10538:
8361:
3969:
262:
19679:
5818:
133:
19664:
19659:
19604:
19540:
19484:
19305:
19292:
19083:
19028:
18980:
18771:
18700:
18565:
18462:
18457:
17411:
17406:
15204:
15153:
13100:
12791:
12599:
12590:
12144:
When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the
12128:
11982:â similar to the QâQ plot, but used much less frequently. This method consists of plotting the points (Ί(
11647:
10820:
6910:
2574:
31:
13005:, illustrates an example of fitting the normal distribution to ranked October rainfalls showing the 90%
12609:. If initially the particle is located at a specific point (that is its probability distribution is the
4740:
4615:{\textstyle \mu ^{8}+28\mu ^{6}\sigma ^{2}+210\mu ^{4}\sigma ^{4}+420\mu ^{2}\sigma ^{6}+105\sigma ^{8}}
19798:
19574:
19393:
19175:
19128:
18997:
18973:
18953:
18796:
18670:
18550:
14825:
14804:
12927:
12537:
12173:
11866:
11683:
7975:
6184:
1540:
848:
683:
233:
12202:
3895:
19803:
19587:
19548:
19422:
19259:
19103:
19048:
18946:
18910:
18781:
18746:
16139:
15184:
15164:
13547:
Numerical approximations for the normal cumulative distribution function and normal quantile function
12502:
12119:
7877:
5772:
5595:
1656:
17719:
15775:
15483:
7478:
7440:
6308:
3771:
19489:
19277:
19043:
19002:
18917:
18871:
18811:
18776:
18665:
18560:
18510:
13278:
13274:
13107:
12415:
10595:
8853:
8484:
8334:
8070:
6805:
1715:
1710:
1599:
1584:
14816:
11877:
8079:
7717:
7516:
6480:
As the number of discrete events increases, the function begins to resemble a normal distribution.
6354:
19788:
19730:
19401:
19188:
19098:
19053:
19038:
18856:
18806:
18801:
18602:
18582:
17220:
16670:
Wichura, Michael J. (1988). "Algorithm AS241: The
Percentage Points of the Normal Distribution".
16049:
14739:
12967:
12893:
12863:
11917:
11818:
11799:
11776:
11619:
8781:
8749:
7779:
7258:
7225:
3401:
3359:
1694:
1565:
655:
18958:
16525:
John, S (1982). "The three parameter two-piece normal family of distributions and its fitting".
15603:
7291:
4626:
3327:
19654:
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19631:
19513:
19409:
19216:
18660:
18640:
18545:
17961:
17714:
17435:
17293:
Products of Random
Variables: Applications to Problems of Physics and to Arithmetical Functions
16016:
15770:
15730:
15415:
15308:
14810:
14104:
13293:
13082:
12389:
12381:{\textstyle {\frac {ab}{a+b}}={\frac {1}{{\frac {1}{a}}+{\frac {1}{b}}}}=(a^{-1}+b^{-1})^{-1}.}
12177:
12145:
8551:
6970:
6272:
6239:
4343:
1589:
17740:
16603:
16333:
10120:
9511:
9065:
8597:
7607:
of 4 iid standard normal variables. These are computed by the numerical method of ray-tracing.
7328:
Heat map of the joint probability density of two functions of two correlated normal variables
6810:
4832:
4112:
1953:
354:{\displaystyle {\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}}
19778:
19735:
19579:
19254:
19108:
19088:
18985:
18555:
18474:
18421:
18115:
Shore, H (2005). "Accurate RMM-Based
Approximations for the CDF of the Normal Distribution".
17338:
15636:
15174:
14846:
14754:
13092:
13010:
12998:
12939:
12921:
12797:
12787:
12563:
12107:
11814:
10779:
10515:
9380:
9176:
9093:
8986:
8813:
7371:
6572:
6565:
5621:
3936:
3744:
1730:
1689:
1594:
1560:
226:
126:
16041:
16008:
15722:
13871:
approximation in the tail to provide a fast computation algorithm with a 16-digit precision.
7407:
7192:
7159:
6980:
3033:
3004:
19828:
19823:
19818:
19813:
19750:
19720:
19599:
19242:
19133:
18736:
18695:
18690:
18587:
17630:
17312:
17152:
16181:"Stat260: Bayesian Modeling and Inference: The Conjugate Prior for the Normal Distribution"
15864:
15664:
15194:
14750:
14735:
14108:
13240:{\displaystyle X={\sqrt {-2\ln U}}\,\cos(2\pi V),\qquad Y={\sqrt {-2\ln U}}\,\sin(2\pi V).}
12983:
12955:
12836:
12803:
12610:
12495:
12102:
11826:
11795:
10816:
9411:
8187:
7062:
6718:
6203:
6199:
4693:
1999:
1720:
1614:
1507:
19033:
18312:
16404:
15872:
8722:
8695:
8591:
6784:
6764:
6744:
6724:
1897:
627:
599:
571:
8:
19762:
19287:
19237:
19211:
19165:
19093:
18905:
18841:
17995:"'Das Fehlergesetz und seine Verallgemeinerungen durch Fechner und Pearson'. A rejoinder"
17879:
14224:
14146:
14120:
12979:
10388:
9405:
7688:
7033:
6915:
6639:
3062:
2952:
1995:
1679:
1621:
1609:
782:
17663:
17316:
16122:
15950:
Winkelbauer, Andreas (2012). "Moments and
Absolute Moments of the Normal Distribution".
14694:
10594:
are independent standard normal random variables, then the sum of their squares has the
8310:
8046:
6577:
3324:
is zero and changes sign), located one standard deviation away from the mean, namely at
19793:
19282:
19063:
19058:
18963:
18900:
18895:
18751:
18741:
18625:
18378:
18363:
18257:
18228:
18190:
18161:
18132:
18103:
18047:
18014:
17978:
17801:
17787:
17732:
17651:
17580:
17545:
17527:
17375:
17263:
17204:
17016:
16687:
16421:
16384:
16297:
16274:
16262:
16137:
Edward L. Melnick and Aaron
Tenenbein, "Misspecifications of the Normal Distribution",
16067:
16042:
15951:
15852:
15391:
15373:
14798:
14790:
13868:
13538:
13523:
12987:
12917:
12622:
12521:
12166:
11693:
10808:
10529:
10273:
10248:
9129:
9102:
8664:
8637:
7092:
6207:
6180:
5352:
3695:
2657:
2605:
1926:
1825:
1666:
1555:
1495:
1472:
1324:
999:
761:
733:
163:
17921:
17904:
17169:
16402:
Basu, D.; Laha, R. G. (1954). "On Some Characterizations of the Normal Distribution".
13406:{\displaystyle X=U{\sqrt {\frac {-2\ln S}{S}}},\qquad Y=V{\sqrt {\frac {-2\ln S}{S}}}}
12862:
of various variables tend to have a normal distribution, that is, they tend to have a
12738:
12569:
Distributions modeled as normal â the normal distribution being the distribution with
12520:
of the mean on the unknown variance, with a hyperparameter specifying the mean of the
12388:
This shows that this factor can be thought of as resulting from a situation where the
10759:
9360:
9156:
7619:
6944:
6491:
5923:
5769:
of the mean of a normal distribution is another normal distribution. Specifically, if
5601:
4797:
3848:
3724:
19691:
19118:
18861:
18791:
18756:
18705:
18434:
18426:
18375:
18323:
18297:
18278:
18271:
18194:
18136:
18061:"Wilhelm Lexis: The Normal Length of Life as an Expression of the "Nature of Things""
17982:
17935:
17808:
17684:
17616:
17597:
17504:
17485:
17468:
17458:
17439:
17392:
17344:
17334:
17298:
17291:
17226:
17216:
17208:
17137:
17118:
17099:
17080:
17020:
17008:
16641:
16609:
16444:
16413:
16376:
16302:
16091:
16053:
16020:
16009:
15844:
15734:
15723:
15701:
15668:
15652:
15642:
15624:
15600:
15460:
15395:
14784:
13878:
13579:
13014:
12905:
12901:
12811:
10826:
10812:
10279:
9039:
9013:
7853:
6537:
fair 6-sided dice to show their convergence to a normal distribution with increasing
6188:
4467:{\textstyle \mu ^{7}+21\mu ^{5}\sigma ^{2}+105\mu ^{3}\sigma ^{4}+105\mu \sigma ^{6}}
2979:
2108:
1725:
1631:
1530:
620:
489:
18367:
18165:
17736:
17584:
17549:
16989:"The Modified-Half-Normal distribution: Properties and an efficient sampling scheme"
15307:
Besides those specifically referenced here, such use is encountered in the works of
13322:
is greater or equal to 1, then the method starts over, otherwise the two quantities
12904:, not like simple interest, and so are multiplicative). Some mathematicians such as
12827:
11664:-dimensional multivariate normal distribution. The variance-covariance structure of
11644:
a rectified version of normal distribution with all the negative elements reset to 0
6866:
6616:
6540:
6212:
18866:
18540:
18479:
18353:
18249:
18218:
18182:
18153:
18124:
18095:
18072:
18039:
18006:
17970:
17916:
17891:
17865:
17838:
17822:
17777:
17724:
17634:
17570:
17537:
17367:
17253:
17196:
17164:
17000:
16679:
16534:
16366:
16292:
16284:
16154:"Kullback Leibler (KL) Distance of Two Normal (Gaussian) Probability Distributions"
15876:
15868:
15836:
15780:
15693:
15383:
15179:
13252:
12562:
Approximately normal laws, for example when such approximation is justified by the
12249:
11689:
10601:
9223:
9203:
8966:
8946:
8493:
8343:
8121:
7957:
7859:
7761:
7668:
7648:
7351:
7331:
7139:
7119:
7013:
6889:
6846:
6697:
6677:
6520:
4957:
4937:
4905:
4885:
4858:
4838:
4720:
4478:
4221:
4016:
3869:
3704:
3307:
3301:
3093:
2636:
2584:
2018:
1550:
1480:
17953:
17323:
Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections
17004:
15784:
11660:
is said to be normal if both its real and imaginary components jointly possess a 2
10049:
Their sum and difference is distributed normally with mean zero and variance two:
8513:
18430:
18422:
Handbook of mathematical functions with formulas, graphs, and mathematical tables
18414:
18202:
17803:
Statistics in Scientific Investigation: Its Basis, Application and Interpretation
17326:
15860:
15660:
15638:
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
15632:
15442:
14801:â similar to the IrwinâHall distribution, but rescaled back into the 0 to 1 range
13070:
13006:
12963:
12533:
12159:
12141:
Either the mean, or the variance, or neither, may be considered a fixed quantity.
12124:
11907:
11791:
11762:
11581:
is multivariate-normally distributed if any linear combination of its components
11568:
9350:{\displaystyle X_{3}={\frac {aX_{1}+bX_{2}-(a+b)\mu }{\sqrt {a^{2}+b^{2}}}}+\mu }
6484:
5766:
1626:
1577:
754:
217:
18240:
Stigler, Stephen M. (1982). "A Modest Proposal: A New Standard for the Normal".
16933:
13073:Ί, which cannot be done analytically. Some approximate methods are described in
11790:
is an analogue of the Gaussian distribution, in the sense that it maximises the
18939:
17639:
Mémoires de l'Académie Royale des Sciences de Paris (Savants étrangers), Tome 6
17429:
17189:"Approximate Incomplete Integrals, Application to Complementary Error Function"
15387:
15312:
14757:
in 1810, consolidating the importance of the normal distribution in statistics.
13078:
13049:
12975:
12163:
11892:
11756:
11732:
11464:
11235:
9547:
4332:{\textstyle \mu ^{6}+15\mu ^{4}\sigma ^{2}+45\mu ^{2}\sigma ^{4}+15\sigma ^{6}}
1641:
564:
478:{\displaystyle \Phi \left({\frac {x-\mu }{\sigma }}\right)={\frac {1}{2}}\left}
17974:
17895:
17782:
17765:
17242:"On the optimal rates of convergence for nonparametric deconvolution problems"
16538:
16371:
16352:
15840:
15687:
15582:
11881:
3768:
is in a particular set, can be calculated by using the fact that the fraction
19871:
19562:
19310:
18597:
18223:
18206:
18043:
17559:"Computer Generation of Random Variables Using the Ratio of Uniform Deviates"
17472:
17258:
17241:
17012:
16417:
16380:
15848:
15822:
15697:
15320:
14830:
14820:
12453:
12409:
11852:
Maximum likelihood § Continuous distribution, continuous parameter space
11697:
10036:
6233:
3644:{\textstyle \varphi ^{(n)}(x)=(-1)^{n}\operatorname {He} _{n}(x)\varphi (x),}
2004:
1514:
16153:
11753:. Several Gaussian processes became popular enough to have their own names:
18402:
17990:
17949:
16630:
16306:
15628:
13424:
The Ratio method is a rejection method. The algorithm proceeds as follows:
13045:
12887:
Certain physiological measurements, such as blood pressure of adult humans.
12866:(after separation on male/female subpopulations), with examples including:
12110:: This is based on the fact that the line in the QâQ plot has the slope of
11979:
11913:
11896:
9062:
are also independent and normally distributed, with zero mean and variance
4927:
1741:
1651:
1535:
18358:
18341:
18128:
18077:
17870:
17853:
17843:
17826:
17728:
17611:
Kruskal, William H.; Stigler, Stephen M. (1997). Spencer, Bruce D. (ed.).
17575:
17558:
17518:
Karney, C. F. F. (2016). "Sampling exactly from the normal distribution".
17050:
16660:
Why Most Published Research Findings Are False, John P. A. Ioannidis, 2005
15880:
10043:
66:
47:
17318:
Theoria motvs corporvm coelestivm in sectionibvs conicis Solem ambientivm
17200:
16208:
14834:
12913:
12769:
11704:, and thus are the analogues of multivariate normal vectors for the case
10804:
3405:
1661:
1502:
1490:
16425:
16288:
12085:{\textstyle \textstyle z_{(k)}=(x_{(k)}-{\hat {\mu }})/{\hat {\sigma }}}
3291:{\textstyle f''(x)={\frac {(x-\mu )^{2}-\sigma ^{2}}{\sigma ^{4}}}f(x).}
18261:
18232:
18107:
18088:
Journal of the Royal Statistical Society. Series C (Applied Statistics)
18051:
18018:
17791:
17655:
17379:
17267:
16691:
16388:
15856:
14746:
14738:
discovered the normal distribution in 1809 as a way to rationalize the
14731:
12884:; presumably the thickness of tree bark also falls under this category;
11813:
is a generalization of the Gaussian distribution which arises from the
11787:
11780:
7752:
4824:
2998:
1519:
1465:
18186:
17499:
Johnson, Norman L.; Kotz, Samuel; Balakrishnan, Narayanaswamy (1995).
17480:
Johnson, Norman L.; Kotz, Samuel; Balakrishnan, Narayanaswamy (1994).
17041:
13537:
There is also some investigation into the connection between the fast
12916:
would be a more appropriate model, in particular for the analysis for
12869:
Measures of size of living tissue (length, height, skin area, weight);
12585:
10522:
8300:{\textstyle X^{2}/\sigma ^{2}\sim \chi _{1}^{2}(\mu ^{2}/\sigma ^{2})}
553:{\displaystyle \mu +\sigma {\sqrt {2}}\operatorname {erf} ^{-1}(2p-1)}
18382:
18273:
The History of Statistics: The Measurement of Uncertainty before 1900
18157:
15756:"Maximum Entropy Autoregressive Conditional Heteroskedasticity Model"
15608:
15598:
15556:
12994:
12606:
11635:
11457:
1820:
18253:
18173:
Shore, H (2012). "Estimating Response Modeling Methodology Models".
18099:
18027:
18010:
17994:
17541:
17371:
17188:
16683:
15672:
13040:
6476:
1826:
Using the Taylor series and Newton's method for the inverse function
18342:"Fast pseudo-random generators for normal and exponential variates"
17068:
17064:
17060:
17056:
16973:
16279:
16088:
Kendall's Advanced Theory of statistics, Vol 2B, Bayesian Inference
16015:(Reprinted. ed.). Cambridge : Cambridge Univ. Press. pp.
15378:
12532:, which is the product of the two distributions just defined, with
4876:
2994:
1646:
726:
648:
213:
17954:"On Lines and Planes of Closest Fit to Systems of Points in Space"
17532:
15956:
13887:
suggested a simple algorithm based on the Taylor series expansion
17431:
The Bell Curve: Intelligence and Class Structure in American Life
17325:] (in Latin). Hambvrgi, Svmtibvs F. Perthes et I. H. Besser.
16631:"Chapter 6: Frequency and Regression Analysis of Hydrologic Data"
13002:
12990:
assigns relative grades based on a normal distribution of scores.
12971:
12855:
12735:. If the initial location is given by a certain density function
12617:
its location is described by a normal distribution with variance
11627:
10756:
are independent normally distributed random variables with means
8628:
7944:{\textstyle \sigma (X)\sim P({\mathcal {N}}(\mu ,\,\sigma ^{2}))}
6192:
15656:
15160:
Normally distributed and uncorrelated does not imply independent
12962:
Many scores are derived from the normal distribution, including
12954:
Fitted cumulative normal distribution to October rainfalls, see
12880:
appendages (hair, claws, nails, teeth) of biological specimens,
12544:
on the variance) and with the same four parameters just defined.
11618:. The multivariate normal distribution is a special case of the
4658:
16011:
Weighing the odds : a course in probability and statistics
15825:(March 1942). "A Characterization of the Normal Distribution".
15210:
15197:â The normal distribution is a member of the family of Tweedie
12950:
12135:
11921:
11696:. These can be viewed as elements of some infinite-dimensional
6965:
2983:
592:
15243:
De Moivre first published his findings in 1733, in a pamphlet
10688:{\displaystyle X_{1}^{2}+\cdots +X_{n}^{2}\sim \chi _{n}^{2}.}
9744:{\textstyle {\mathcal {N}}(m_{\alpha },\sigma _{\alpha }^{2})}
7751:. That is, the family of normal distributions is closed under
16987:
Sun, Jingchao; Kong, Maiying; Pal, Subhadip (June 22, 2021).
13542:
to turn arbitrary data sets into a normally distributed data.
13247:
will both have the standard normal distribution, and will be
13035:
11543:
11530:
of multiple independent or correlated normal variables, is a
4210:{\textstyle \mu ^{5}+10\mu ^{3}\sigma ^{2}+15\mu \sigma ^{4}}
17115:
The Normal Distribution: Characterizations with Applications
12540:
over the variance, and a normal distribution over the mean,
9501:{\textstyle X_{k}\sim {\mathcal {N}}(m_{k},\sigma _{k}^{2})}
8983:
are independent normal deviates with zero mean and variance
4663:
17613:
Normative Terminology: 'Normal' in Statistics and Elsewhere
15357:
Norton, Matthew; Khokhlov, Valentyn; Uryasev, Stan (2019).
13811:) is the standard normal probability density function, and
11779:
is an abstract mathematical construction that represents a
10276:. This distribution is symmetric around zero, unbounded at
2987:
2385:
2317:
2249:
2179:
159:
17635:"Mémoire sur la probabilité des causes par les événements"
17153:"Rational Chebyshev Approximations for the Error Function"
14813:â method used to separate mixtures of normal distributions
13277:
with two degrees of freedom, which is an easily generated
13025:
12180:. The resulting analysis is similar to the basic cases of
17615:. Statistics and Public Policy. Oxford University Press.
17051:"Earliest Known Uses of Some of the Words of Mathematics"
16263:"A method to integrate and classify normal distributions"
16209:"Expectation of the maximum of gaussian random variables"
13304:
are drawn from the uniform (â1,1) distribution, and then
12943:
12488:
10450:{\textstyle X_{1}/X_{2}\sim \operatorname {Cauchy} (0,1)}
8036:{\textstyle {\left|X\right|\sim N_{f}(\mu ,\sigma ^{2})}}
7611:
17042:"Earliest Uses of Symbols in Probability and Statistics"
11650:
deals with the complex normal vectors. A complex vector
11606:
has a (univariate) normal distribution. The variance of
6941:
is approximately normal with mean 0 and variance 1 when
17594:
Handbook of Statistical Distributions with Applications
17498:
17479:
16783:
16704:
16504:"On three characterisations of the normal distribution"
16128:, 21.6:"Individually Gaussian Versus Jointly Gaussian".
15290:"My custom of terming the curve the GaussâLaplacian or
10044:
Operations on two independent standard normal variables
5920:, then the posterior distribution for the estimator of
4101:{\textstyle \mu ^{4}+6\mu ^{2}\sigma ^{2}+3\sigma ^{4}}
18392:"Better Approximations to Cumulative Normal Functions"
15937:
Probability, Random Variables and Stochastic Processes
15207:â the Normal distribution applied to a circular domain
15079:
15032:
14881:
14849:
14697:
14252:
14227:
14178:
14149:
14123:
13085:
to compute random variates of the normal distribution.
12741:
12631:
12418:
12268:
12205:
12007:
12006:
11473:
11180:
11121:
10829:
10782:
10762:
10703:
10604:
10541:
10466:
10397:
10311:
10282:
10251:
10170:
10123:
10055:
9917:
9757:
9699:
9555:
9514:
9443:
9414:
9383:
9363:
9226:
9206:
9179:
9159:
9132:
9105:
9068:
9042:
9016:
8989:
8969:
8949:
8896:
8856:
8816:
8784:
8752:
8725:
8698:
8667:
8640:
8600:
8554:
8516:
8496:
8346:
8313:
8222:
8190:
8130:
8082:
8049:
7984:
7960:
7886:
7862:
7788:
7764:
7720:
7691:
7671:
7651:
7622:
7559:
7519:
7481:
7443:
7410:
7374:
7354:
7334:
7294:
7261:
7228:
7195:
7162:
7142:
7122:
7095:
7065:
7036:
7016:
6983:
6947:
6918:
6892:
6869:
6849:
6813:
6787:
6767:
6747:
6727:
6700:
6680:
6642:
6619:
6580:
6543:
6523:
6494:
6399:
6357:
6311:
6275:
6242:
6215:
5926:
5866:
5821:
5775:
5689:
5624:
5604:
5058:
4991:
4960:
4940:
4908:
4888:
4861:
4841:
4800:
4743:
4723:
4696:
4629:
4507:
4481:
4382:
4346:
4250:
4224:
4151:
4115:
4045:
4019:
3972:
3939:
3898:
3872:
3851:
3774:
3747:
3727:
3707:
3657:
3556:
3479:
3419:
3362:
3330:
3310:
3201:
3186:{\textstyle f'(x)=-{\frac {x-\mu }{\sigma ^{2}}}f(x).}
3119:
3096:
3065:
3036:
3007:
2955:
2660:
2639:
2608:
2587:
2138:
2111:
2042:
2021:
1956:
1929:
1900:
1836:
1047:
18433:: National Bureau of Standards. New York, NY: Dover.
17827:"The Ziggurat Method for Generating Random Variables"
15356:
13893:
13588:
13328:
13135:
11538:
11257:
10855:
10624:
9246:
8370:
6179:
The family of normal distributions not only forms an
5946:
5651:
5361:
5125:
4855:
can be at most a quadratic polynomial, and therefore
3701:
The probability that a normally distributed variable
1334:
1141:
1009:
932:
858:
837:{\displaystyle {\frac {1}{2}}\log(2\pi e\sigma ^{2})}
792:
764:
736:
693:
658:
630:
602:
574:
499:
375:
272:
236:
173:
136:
82:
18509:
16638:
Drainage Principles and Applications, Publication 16
15445:, The British Journal for the Philosophy of Science.
15230:
For example, this algorithm is given in the article
14780:
14214:{\textstyle \left(\approx 1.1\times 10^{-16}\right)}
12839:, with superimposed best-fitting normal distribution
12598:
Probability density function of a ground state in a
11686:
describes the case of normally distributed matrices.
17357:
15720:
15518:
12473:
10532:
of independent normal deviates is a normal deviate.
10523:
Operations on multiple independent normal variables
10107:{\textstyle X_{1}\pm X_{2}\sim {\mathcal {N}}(0,2)}
6461:. Note that there is no assumption of independence.
1810:
205:{\displaystyle \sigma ^{2}\in \mathbb {R} _{>0}}
18270:
18144:Shore, H (2011). "Response Modeling Methodology".
17800:
17290:
16143:, volume 36, number 4 November 1982, pages 372â373
15144:
15018:
14867:
14819:â on the occurrence of the normal distribution in
14709:
14683:
14239:
14213:
14165:
14135:
14077:
13795:
13421:are independent, standard normal random variables.
13405:
13239:
12756:
12727:
12444:
12380:
12240:
12084:
11522:
11458:Operations on multiple correlated normal variables
11446:
11225:
11166:
11105:
10841:
10819:. The ratio of these two quantities will have the
10795:
10768:
10748:
10687:
10610:
10586:
10506:
10449:
10376:
10294:
10264:
10237:
10152:
10106:
10027:
9903:
9743:
9685:
9538:
9500:
9426:
9396:
9369:
9349:
9232:
9212:
9192:
9165:
9145:
9118:
9084:
9054:
9028:
9002:
8975:
8955:
8932:
8882:
8842:
8802:
8770:
8738:
8711:
8680:
8653:
8616:
8582:
8534:
8502:
8475:
8352:
8325:
8299:
8204:
8173:
8112:
8061:
8035:
7966:
7943:
7868:
7841:
7770:
7743:
7706:
7677:
7657:
7637:
7599:
7541:
7505:
7467:
7429:
7396:
7360:
7340:
7316:
7280:
7247:
7214:
7181:
7148:
7128:
7108:
7077:
7051:
7022:
7002:
6953:
6933:
6898:
6878:
6855:
6835:
6793:
6773:
6753:
6733:
6706:
6686:
6666:
6628:
6601:
6552:
6529:
6509:
6453:
6385:
6343:
6294:
6261:
6224:
6169:
5932:
5912:
5852:
5807:
5755:
5637:
5610:
5584:
5343:
5112:{\textstyle X_{2}\sim N(\mu _{2},\sigma _{2}^{2})}
5111:
5045:{\textstyle X_{1}\sim N(\mu _{1},\sigma _{1}^{2})}
5044:
4966:
4946:
4914:
4894:
4867:
4847:
4815:
4786:
4729:
4709:
4645:
4614:
4487:
4466:
4362:
4331:
4230:
4209:
4131:
4100:
4025:
4004:
3952:
3924:
3878:
3857:
3806:
3760:
3733:
3713:
3682:
3643:
3536:
3463:
3383:
3348:
3316:
3290:
3185:
3102:
3080:
3051:
3022:
2970:
2673:
2645:
2621:
2593:
2169:
2123:
2096:
2027:
1989:
1978:
1942:
1915:
1886:
1821:Recursive computation with Taylor series expansion
1448:
1312:
1119:
987:
910:
836:
770:
742:
714:
671:
636:
608:
580:
552:
477:
353:
250:
204:
150:
114:
18117:Communications in Statistics â Theory and Methods
17288:
16993:Communications in Statistics â Theory and Methods
16980:
16527:Communications in Statistics â Theory and Methods
16475:
16048:(Reprint ed.). Chichester : Wiley. pp.
11856:Gaussian function § Estimation of parameters
5913:{\textstyle \mu \sim N(\mu _{0},\sigma _{0}^{2})}
30:"Bell curve" redirects here. For other uses, see
19869:
17556:
17423:
16974:"Earliest Uses... (Entry Standard Normal Curve)"
16716:
16608:. Cambridge University Press. pp. 592â593.
15623:
15555:Scott, Clayton; Nowak, Robert (August 7, 2003).
14837:, which uses the normal distribution as a kernel
13057:The most straightforward method is based on the
9546:are normal distributions, then their normalized
6407:
2541:
2481:
2424:
2356:
2288:
2217:
2097:{\textstyle p=F(\mu +n\sigma )-F(\mu -n\sigma )}
1805:
115:{\displaystyle {\mathcal {N}}(\mu ,\sigma ^{2})}
19903:Location-scale family probability distributions
18058:
17557:Kinderman, Albert J.; Monahan, John F. (1977).
17094:Bernardo, José M.; Smith, Adrian F. M. (2000).
16353:"A Characterization of the Normal Distribution"
16106:
16104:
15328:
7842:{\textstyle e^{X}\sim \ln(N(\mu ,\sigma ^{2}))}
3537:{\textstyle \varphi ''(x)=(x^{2}-1)\varphi (x)}
1790:
988:{\displaystyle \exp(i\mu t-\sigma ^{2}t^{2}/2)}
18059:Rohrbasser, Jean-Marc; VĂ©ron, Jacques (2003).
17821:
17681:Asymptotics in Statistics: Some Basic Concepts
17610:
17591:
16960:
16738:
16578:
16566:
16554:
16234:"Normal Approximation to Poisson Distribution"
16040:Smith, José M. Bernardo; Adrian F. M. (2000).
15821:
12553:
11467:of a normal vector, i.e. a quadratic function
9240:are arbitrary real numbers, then the variable
9153:are two independent normal deviates with mean
8629:Operations on two independent normal variables
1795:
911:{\displaystyle \exp(\mu t+\sigma ^{2}t^{2}/2)}
18495:
18207:"Mathematical Statistics in the Early States"
17930:Patel, Jagdish K.; Read, Campbell B. (1996).
17678:
17501:Continuous Univariate Distributions, Volume 2
17482:Continuous Univariate Distributions, Volume 1
17131:
17093:
17069:"Error, law of error, theory of errors, etc."
16807:De Moivre, Abraham (1733), Corollary I â see
16795:
15542:
15429:
9404:. It follows that the normal distribution is
8850:will also be normally distributed, with mean
7600:{\textstyle \sum _{i=1}^{4}\vert x_{i}\vert }
6488:Comparison of probability density functions,
6202:, the family of normal distributions forms a
4659:Fourier transform and characteristic function
2943:
1766:
18313:"De Moivre on the Law of Normal Probability"
17391:. New York, NY: John Wiley & Sons, Inc.
17074:
16196:
16101:
16072:: CS1 maint: multiple names: authors list (
15478:
15476:
15454:
15190:Sum of normally distributed random variables
12800:, associated with binary response variables;
12776:and the normal probability density function.
12605:The position of a particle that experiences
12136:Bayesian analysis of the normal distribution
11548:
9533:
9521:
8933:{\textstyle \sigma _{1}^{2}+\sigma _{2}^{2}}
8076:The absolute value of normalized residuals,
7594:
7581:
6966:Operations and functions of normal variables
2575:Quantile function § Normal distribution
2170:{\textstyle {\text{or }}1{\text{ in }}(1-p)}
18412:
17177:
17075:Amari, Shun-ichi; Nagaoka, Hiroshi (2000).
15994:
15949:
15455:Jorge, Nocedal; Stephan, J. Wright (2006).
13552:
11845:
10274:modified Bessel function of the second kind
8174:{\textstyle |X-\mu |/\sigma \sim \chi _{1}}
5355:between the same distributions is equal to
4668:
18502:
18488:
17763:
17289:Galambos, Janos; Simonelli, Italo (2004).
17132:Casella, George; Berger, Roger L. (2001).
16986:
16628:
16490:
15894:
15892:
15890:
15554:
15245:Approximatio ad Summam Terminorum Binomii
13427:Generate two independent uniform deviates
13065:is distributed uniformly on (0,1), then Ί(
13036:Generating values from normal distribution
12931:about the mean and variance of the errors.
11798:. This distribution is different from the
11544:Infinite divisibility and Cramér's theorem
11523:{\textstyle q=\sum x_{i}^{2}+\sum x_{j}+c}
10507:{\textstyle {\sqrt {X_{1}^{2}+X_{2}^{2}}}}
10377:{\textstyle \phi _{Z}(t)=(1+t^{2})^{-1/2}}
7685:, is also normally distributed, with mean
6454:{\textstyle E\leq \sigma {\sqrt {2\ln n}}}
1773:
1759:
18413:Zelen, Marvin; Severo, Norman C. (1964).
18357:
18346:ACM Transactions on Mathematical Software
18222:
18076:
17929:
17920:
17869:
17851:
17842:
17798:
17781:
17718:
17707:ACM Transactions on Mathematical Software
17574:
17563:ACM Transactions on Mathematical Software
17531:
17520:ACM Transactions on Mathematical Software
17257:
17215:
17178:Cover, Thomas M.; Thomas, Joy A. (2006).
17168:
17048:
17039:
16550:
16548:
16463:
16370:
16296:
16278:
15955:
15922:
15898:
15806:
15774:
15721:Cover, Thomas M.; Thomas, Joy A. (2006).
15686:Vaart, A. W. van der (October 13, 1998).
15530:
15473:
15377:
13884:
13281:corresponding to the quantity â2 ln(
13212:
13161:
11721:is said to be normal if for any constant
10238:{\textstyle f_{Z}(z)=\pi ^{-1}K_{0}(|z|)}
9635:
9569:
7924:
4664:Moment- and cumulant-generating functions
244:
189:
144:
16930:Probability Theory: The Logic of Science
16605:Probability Theory: The Logic of Science
16518:
16401:
16006:
15753:
14745:
14730:
14143:with a maximum relative error less than
14117:proposes the following approximation of
13251:. This formulation arises because for a
13122:on (0,1). Then the two random variables
13039:
13030:
12949:
12826:
12780:
12584:
12188:
11840:
11614:symmetric positive-definite matrix
8340:The log-likelihood of a normal variable
6969:
6483:
6475:
6471:
6466:
3464:{\textstyle \varphi '(x)=-x\varphi (x).}
2003:
1887:{\textstyle \Phi (x,x_{0},\Phi (x_{0}))}
18336:
18291:
18268:
18239:
18201:
18025:
17989:
17948:
17902:
17878:
17764:Lukacs, Eugene; King, Edgar P. (1954).
17700:"A fast normal random number generator"
17679:Le Cam, Lucien; Lo Yang, Grace (2000).
17661:
17629:
16916:
16904:
16892:
16880:
16868:
16856:
16820:
16772:
16760:
16669:
16508:Probability and Mathematical Statistics
16443:(2nd ed.). Springer. p. 199.
16438:
16179:Jordan, Michael I. (February 8, 2010).
15887:
15507:
15459:(2nd ed.). Springer. p. 249.
15295:
14769:
13089:An easy-to-program approximate approach
13026:Methodological problems and peer review
13013:. The rainfall data are represented by
12846:
12169:may be placed on the unknown variables.
11886:
11622:. As such, its iso-density loci in the
9357:is also normally distributed with mean
3825:List of integrals of Gaussian functions
3683:{\textstyle \operatorname {He} _{n}(x)}
715:{\displaystyle \sigma {\sqrt {2/\pi }}}
14:
19870:
18310:
17905:"Accuracy in random number generation"
17517:
17452:
17274:
16808:
16784:Johnson, Kotz & Balakrishnan (1994
16749:
16705:Johnson, Kotz & Balakrishnan (1995
16601:
16590:
16545:
16350:
16256:
16254:
16178:
15316:
15266:
13867:combines Hart's algorithm 5666 with a
12489:With unknown mean and unknown variance
12478:
11226:{\textstyle Y_{1},Y_{2},\ldots ,Y_{m}}
11167:{\textstyle X_{1},X_{2},\ldots ,X_{n}}
10749:{\textstyle X_{1},X_{2},\ldots ,X_{n}}
10587:{\textstyle X_{1},X_{2},\ldots ,X_{n}}
7612:Operations on a single normal variable
4673:
4005:{\textstyle \mu ^{3}+3\mu \sigma ^{2}}
3090:The area bounded by the curve and the
18483:
18374:
18172:
18143:
18114:
18085:
18028:"Notes on the History of Correlation"
17770:The Annals of Mathematical Statistics
17753:Annales de DĂ©mographie Internationale
17750:
17333:
17311:
16844:
16832:
16501:
16486:
16484:
16358:The Annals of Mathematical Statistics
16331:
16039:
15754:Park, Sung Y.; Bera, Anil K. (2009).
15685:
15599:
15581:. Tel Aviv University. Archived from
15573:
15350:
15324:
15278:
13512:, otherwise start over the algorithm.
12172:An additional set of cases occurs in
6236:with respect to the (±1)-connections
5853:{\textstyle \sim N(\mu ,\sigma ^{2})}
19852:
18389:
17854:"Evaluating the Normal Distribution"
17697:
17643:Translated by Stephen M. Stigler in
17386:
17284:. London, UK: Richard Clay and Sons.
17150:
17112:
16727:
16524:
16319:
16125:The Multivariate Normal Distribution
16110:
15982:
15970:
15934:
15443:Why are Normal Distributions Normal?
13874:
13864:
13856:
13110:uses two independent random numbers
13074:
12822:
12794:distributions are involved, such as
11878:Standard deviation § Estimation
8692:normal random variables, with means
8333:, the distribution is called simply
2568:
1923:from a Taylor series solution using
151:{\displaystyle \mu \in \mathbb {R} }
17932:Handbook of the Normal Distribution
17766:"A Property of Normal Distribution"
17665:Théorie analytique des probabilités
17387:Hart, John F.; et al. (1968).
17239:
17186:
16260:
16251:
15910:
14114:
13095:is as follows: generate 12 uniform
12182:independent identically distributed
8214:noncentral chi-squared distribution
4787:{\textstyle \phi _{X}(t)=\exp Q(t)}
4683:
3814:has a standard normal distribution.
3110:-axis is unity (i.e. equal to one).
24:
18416:Probability Functions (chapter 26)
17671:Analytical theory of probabilities
17592:Krishnamoorthy, Kalimuthu (2006).
16481:
15519:Halperin, Hartley & Hoel (1965
15064:
15033:
14973:
14859:
14263:
14130:
13894:
13589:
12691:
12681:
12638:
12634:
12580:
12483:
11901:
11871:
11668:is described by two matrices: the
11563:describes the Gaussian law in the
11539:Operations on the density function
10811:, which can be demonstrated using
10084:
9702:
9459:
7910:
7553:Probability density of a function
7089:Probability density of a function
6977:Probability density of a function
6843:is approximately normal with mean
6741:is approximately normal with mean
6277:
6244:
5987:
5654:
5135:
5132:
4678:
1957:
1901:
1862:
1837:
1012:
376:
85:
25:
19914:
18450:
17922:10.1090/S0025-5718-1985-0804945-X
17225:. American Mathematical Society.
17170:10.1090/S0025-5718-1969-0247736-4
16945:Peirce, Charles S. (c. 1909 MS),
15828:Annals of Mathematical Statistics
15420:, Gale Encyclopedia of Psychology
14841:Modified half-normal distribution
12530:normal-inverse-gamma distribution
12241:{\textstyle {\frac {ay+bz}{a+b}}}
10387:Their ratio follows the standard
8490:The distribution of the variable
5598:for a normal distribution w.r.t.
4794:in a neighborhood of zero, where
3925:{\textstyle \mu ^{2}+\sigma ^{2}}
2949:It is symmetric around the point
1815:
251:{\displaystyle x\in \mathbb {R} }
19893:Exponential family distributions
19851:
19842:
19841:
18318:. In Smith, David Eugene (ed.).
17343:(first ed.). W. W. Norton.
14783:
13879:Rational Chebyshev Approximation
12474:Sum of differences from the mean
11561:multivariate normal distribution
9092:. This is a special case of the
5808:{\textstyle x_{1},\ldots ,x_{n}}
3304:(where the second derivative of
1811:Cumulative distribution function
1479:
65:
63:Cumulative distribution function
46:
17858:Journal of Statistical Software
17831:Journal of Statistical Software
17077:Methods of Information Geometry
17055:In particular, the entries for
16966:
16954:
16939:
16922:
16910:
16898:
16886:
16874:
16862:
16850:
16838:
16826:
16814:
16801:
16789:
16777:
16766:
16754:
16743:
16732:
16721:
16710:
16698:
16663:
16654:
16622:
16595:
16584:
16572:
16560:
16495:
16469:
16457:
16432:
16395:
16344:
16325:
16313:
16226:
16201:
16190:
16172:
16146:
16131:
16116:
16080:
16033:
16000:
15988:
15976:
15964:
15943:
15928:
15916:
15904:
15815:
15747:
15729:. John Wiley and Sons. p.
15714:
15679:
15617:
15592:
15576:"Q Function and Error Function"
15567:
15548:
15536:
15301:
15284:
15271:
15237:
15224:
15213:â using the normal distribution
13755:
13563:> 0 with the absolute error
13367:
13186:
12516:This suggests that we create a
12412:, so it is not surprising that
11834:generalized normal distribution
11642:Rectified Gaussian distribution
10807:is independent from the sample
7506:{\textstyle \sigma _{y}^{2}=20}
7468:{\textstyle \sigma _{x}^{2}=10}
6344:{\textstyle X_{1},\dots ,X_{n}}
5645:is diagonal and takes the form
4690:If the characteristic function
3807:{\textstyle Z=(X-\mu )/\sigma }
1990:Standard deviation and coverage
18475:Normal distribution calculator
17662:Laplace, Pierre-Simon (1812).
17180:Elements of Information Theory
16717:Kinderman & Monahan (1977)
16629:Oosterbaan, Roland J. (1994).
16476:Galambos & Simonelli (2004
16441:Testing Statistical Hypotheses
15725:Elements of Information Theory
15692:. Cambridge University Press.
15604:"Normal Distribution Function"
15524:
15512:
15501:
15448:
15435:
15423:
15409:
15170:Reciprocal normal distribution
15124:
15112:
15048:
15036:
14968:
14940:
14891:
14885:
14862:
14850:
14726:
13931:
13925:
13903:
13897:
13749:
13743:
13619:
13613:
13598:
13592:
13231:
13219:
13180:
13168:
13059:probability integral transform
13001:. The blue picture, made with
12892:In finance, in particular the
12831:Histogram of sepal widths for
12806:, associated with rare events;
12751:
12745:
12722:
12710:
12662:
12650:
12593:has the Gaussian distribution.
12573:for a given mean and variance.
12528:This leads immediately to the
12468:
12445:{\textstyle {\frac {ab}{a+b}}}
12363:
12330:
12193:
12075:
12061:
12055:
12041:
12035:
12027:
12019:
12013:
11860:
11064:
11037:
11019:
10992:
10981:
10969:
10949:
10917:
10444:
10432:
10354:
10334:
10328:
10322:
10232:
10228:
10220:
10216:
10187:
10181:
10101:
10089:
10004:
9992:
9880:
9868:
9817:
9805:
9738:
9707:
9632:
9626:
9602:
9596:
9495:
9464:
9307:
9295:
8883:{\textstyle \mu _{1}+\mu _{2}}
8568:
8555:
8529:
8517:
8386:
8380:
8294:
8266:
8146:
8132:
8098:
8084:
8029:
8010:
7938:
7935:
7915:
7905:
7896:
7890:
7836:
7833:
7814:
7808:
6928:
6922:
6830:
6824:
6661:
6649:
6596:
6584:
6504:
6498:
6426:
6403:
6380:
6361:
6287:
6281:
6254:
6248:
6051:
5907:
5876:
5847:
5828:
5678:
5659:
5530:
5503:
5398:
5372:
5203:
5176:
5167:
5141:
5106:
5075:
5039:
5008:
4810:
4804:
4781:
4775:
4760:
4754:
3793:
3781:
3677:
3671:
3635:
3629:
3623:
3617:
3595:
3585:
3579:
3573:
3568:
3562:
3531:
3525:
3519:
3500:
3494:
3488:
3455:
3449:
3434:
3428:
3282:
3276:
3238:
3225:
3216:
3210:
3177:
3171:
3134:
3128:
2978:which is at the same time the
2164:
2152:
2091:
2076:
2067:
2052:
1973:
1960:
1910:
1904:
1881:
1878:
1865:
1840:
1785:
1546:Collectively exhaustive events
1229:
1202:
1106:
1090:
1036:
1017:
982:
939:
905:
865:
831:
809:
547:
532:
322:
309:
109:
90:
13:
1:
19888:Conjugate prior distributions
17683:(second ed.). Springer.
17049:Aldrich, John; Miller, Jeff.
17040:Aldrich, John; Miller, Jeff.
17005:10.1080/03610926.2021.1934700
16636:. In Ritzema, Henk P. (ed.).
16334:"Normal Product Distribution"
15785:10.1016/j.jeconom.2008.12.014
15574:Barak, Ohad (April 6, 2006).
15366:Annals of Operations Research
15338:
15329:Rohrbasser & VĂ©ron (2003)
15199:exponential dispersion models
13555:give the approximation for Ί(
13019:cumulative frequency analysis
12559:Exactly normal distributions;
11692:are the normally distributed
11630:and in the case of arbitrary
11553:
8544:truncated normal distribution
8113:{\textstyle |X-\mu |/\sigma }
7744:{\textstyle a^{2}\sigma ^{2}}
7542:{\textstyle \rho _{xy}=0.495}
6386:{\textstyle N(0,\sigma ^{2})}
6351:are distributed according to
2938:
1806:Alternative parameterizations
18320:A Source Book in Mathematics
18296:. Harvard University Press.
18292:Stigler, Stephen M. (1999).
18277:. Harvard University Press.
18269:Stigler, Stephen M. (1986).
17057:"bell-shaped and bell curve"
16961:Kruskal & Stigler (1997)
16739:Marsaglia & Tsang (2000)
15939:(4th ed.). p. 148.
15372:(1â2). Springer: 1281â1315.
15343:
12966:(percentiles or quantiles),
12400:add directly, so to combine
12155:cases need to be considered.
11968: + 1 â 2
11058:
11013:
10870:
8803:{\textstyle \sigma _{2}^{2}}
8771:{\textstyle \sigma _{1}^{2}}
8362:probability density function
8216:with one degree of freedom:
8124:with one degree of freedom:
7281:{\textstyle \sigma _{y}=0.2}
7248:{\textstyle \sigma _{x}=0.1}
3384:{\textstyle x=\mu +\sigma .}
1791:Standard normal distribution
54:standard normal distribution
44:Probability density function
7:
18463:Encyclopedia of Mathematics
17934:(2nd ed.). CRC Press.
17674:]. Paris, Ve. Courcier.
17455:Problems of Relative Growth
17412:Encyclopedia of Mathematics
17079:. Oxford University Press.
15205:Wrapped normal distribution
14776:
13497:and start over from step 1;
13279:exponential random variable
12764:, then the density at time
12600:quantum harmonic oscillator
12591:quantum harmonic oscillator
12554:Occurrence and applications
12098:D'Agostino's K-squared test
11783:of the normal distribution.
11648:Complex normal distribution
7317:{\textstyle \rho _{xy}=0.8}
4985:of one normal distribution
4983:KullbackâLeibler divergence
4646:{\textstyle 105\sigma ^{8}}
3349:{\textstyle x=\mu -\sigma }
1800:
1796:General normal distribution
1132:KullbackâLeibler divergence
672:{\displaystyle \sigma ^{2}}
32:Bell curve (disambiguation)
10:
19919:
19675:Wrapped asymmetric Laplace
18646:Extended negative binomial
17909:Mathematics of Computation
17852:Marsaglia, George (2004).
17596:. Chapman & Hall/CRC.
17453:Huxley, Julian S. (1932).
17157:Mathematics of Computation
17113:Bryc, Wlodzimierz (1995).
17031:
16796:Le Cam & Lo Yang (2000
16213:Mathematics Stack Exchange
16197:Amari & Nagaoka (2000)
15543:Bernardo & Smith (2000
15430:Casella & Berger (2001
15388:10.1007/s10479-019-03373-1
14826:Full width at half maximum
14721:
12882:in the direction of growth
12538:inverse gamma distribution
12174:Bayesian linear regression
11905:
11890:
11882:Variance § Estimation
11875:
11867:Standard error of the mean
11864:
11849:
11769:OrnsteinâUhlenbeck process
11684:Matrix normal distribution
8594:with location 0 and scale
8583:{\textstyle (X-\mu )^{-2}}
8510:restricted to an interval
7976:folded normal distribution
7878:logit-normally distributed
6563:
6295:{\textstyle \nabla ^{(m)}}
6262:{\textstyle \nabla ^{(e)}}
6185:natural exponential family
6183:(EF), but in fact forms a
4363:{\textstyle 15\sigma ^{6}}
3822:
3818:
3400:Its density is infinitely
2944:Symmetries and derivatives
2572:
1993:
29:
19837:
19771:
19729:
19630:
19466:
19444:
19435:
19334:Generalized extreme value
19319:
19154:
19114:Relativistic BreitâWigner
18830:
18727:
18718:
18611:
18531:
18522:
18511:Probability distributions
18311:Walker, Helen M. (1985).
18242:The American Statistician
17975:10.1080/14786440109462720
17896:10.1080/14786446008642818
17825:; Tsang, Wai Wan (2000).
17360:The American Statistician
17151:Cody, William J. (1969).
17136:(2nd ed.). Duxbury.
16602:Jaynes, Edwin T. (2003).
16539:10.1080/03610928208828279
16140:The American Statistician
15633:"Chapter 26, eqn 26.2.12"
15185:Sub-Gaussian distribution
15165:Ratio normal distribution
14761:
13553:Zelen & Severo (1964)
12798:Binomial random variables
12503:sum of squared deviations
11942:), where plotting points
11549:The KacâBernstein theorem
10153:{\textstyle Z=X_{1}X_{2}}
9693:is a normal distribution
9539:{\textstyle k\in \{0,1\}}
9085:{\textstyle 2\sigma ^{2}}
8617:{\textstyle \sigma ^{-2}}
8360:is simply the log of its
6836:{\textstyle \chi ^{2}(k)}
5596:Fisher information matrix
4132:{\textstyle 3\sigma ^{4}}
3473:Its second derivative is
3195:Its second derivative is
2630:
1979:{\textstyle \Phi (x_{0})}
1328:
1323:
1135:
1130:
1003:
998:
926:
921:
852:
847:
786:
781:
758:
753:
730:
725:
687:
682:
652:
647:
624:
619:
596:
591:
568:
563:
493:
488:
369:
364:
266:
261:
230:
225:
130:
125:
76:
73:
61:
42:
19883:Continuous distributions
18211:The Annals of Statistics
17799:McPherson, Glen (1990).
17698:Leva, Joseph L. (1992).
17631:Laplace, Pierre-Simon de
17424:Herrnstein, Richard J.;
17275:Galton, Francis (1889).
17246:The Annals of Statistics
16491:Lukacs & King (1954)
16007:Williams, David (2001).
15995:Cover & Thomas (2006
15698:10.1017/cbo9780511802256
15217:
14868:{\textstyle (0,\infty )}
13478:and terminate algorithm;
13275:chi-squared distribution
12968:normal curve equivalents
12804:Poisson random variables
11846:Estimation of parameters
11819:Kaniadakis distributions
11620:elliptical distributions
10821:Student's t-distribution
10796:{\textstyle \sigma ^{2}}
10596:chi-squared distribution
9397:{\textstyle \sigma ^{2}}
9193:{\textstyle \sigma ^{2}}
9003:{\textstyle \sigma ^{2}}
8843:{\textstyle X_{1}+X_{2}}
8071:half-normal distribution
7397:{\textstyle \mu _{x}=-2}
7116:of two normal variables
6911:Student's t-distribution
6806:chi-squared distribution
6714:not too close to 0 or 1.
5638:{\textstyle \sigma ^{2}}
4717:of some random variable
4669:Stein operator and class
3953:{\textstyle \sigma ^{2}}
3761:{\textstyle \sigma ^{2}}
3413:Its first derivative is
3113:Its first derivative is
1716:Law of total probability
1711:Conditional independence
1600:Exponential distribution
1585:Probability distribution
27:Probability distribution
19329:Generalized chi-squared
19273:Normal-inverse Gaussian
18401:: 70â76. Archived from
18294:Statistics on the Table
17903:Monahan, J. F. (1985).
17783:10.1214/aoms/1177728796
17389:Computer Approximations
17313:Gauss, Carolo Friderico
17222:The Doctrine of Chances
17061:"normal (distribution)"
16439:Lehmann, E. L. (1997).
16372:10.1214/aoms/1177731647
16351:Lukacs, Eugene (1942).
15841:10.1214/AOMS/1177731647
15763:Journal of Econometrics
15232:Bc programming language
15154:FoxâWright Psi function
14740:method of least squares
14221:in absolute value: for
13048:, a device invented by
12864:log-normal distribution
12178:regression coefficients
12129:KolmogorovâSmirnov test
11918:normal probability plot
11800:Gaussian q-distribution
11777:Gaussian q-distribution
11672:matrix Î, and the
10304:characteristic function
7645:, for any real numbers
7430:{\textstyle \mu _{y}=5}
7215:{\textstyle \mu _{y}=2}
7182:{\textstyle \mu _{x}=1}
7003:{\textstyle \cos x^{2}}
4934:. The requirement that
3052:{\textstyle x>\mu ,}
3023:{\textstyle x<\mu ,}
1695:Conditional probability
19641:Univariate (circular)
19202:Generalized hyperbolic
18631:ConwayâMaxwellâPoisson
18621:Beta negative binomial
18224:10.1214/aos/1176344123
18044:10.1093/biomet/13.1.25
18026:Pearson, Karl (1920).
17962:Philosophical Magazine
17884:Philosophical Magazine
17297:. Marcel Dekker, Inc.
17259:10.1214/aos/1176348248
17240:Fan, Jianqing (1991).
17182:. John Wiley and Sons.
16464:Patel & Read (1996
16261:Das, Abhranil (2021).
15935:Papoulis, Athanasios.
15923:Patel & Read (1996
15899:Patel & Read (1996
15457:Numerical Optimization
15146:
15020:
14869:
14811:Bhattacharyya distance
14805:BehrensâFisher problem
14774:
14758:
14743:
14711:
14685:
14241:
14215:
14167:
14137:
14105:GNU Scientific Library
14079:
13797:
13407:
13294:Marsaglia polar method
13241:
13053:
12958:
12910:log-Levy distributions
12851:
12840:
12758:
12729:
12621:, which satisfies the
12594:
12589:The ground state of a
12446:
12382:
12242:
12127:(an adaptation of the
12086:
11811:-Gaussian distribution
11532:generalized chi-square
11524:
11448:
11227:
11168:
11107:
10843:
10797:
10770:
10750:
10689:
10612:
10588:
10508:
10451:
10378:
10296:
10266:
10239:
10164:with density function
10154:
10108:
10029:
9905:
9745:
9687:
9540:
9502:
9428:
9427:{\textstyle \alpha =2}
9398:
9371:
9351:
9234:
9214:
9194:
9167:
9147:
9120:
9086:
9056:
9030:
9004:
8977:
8957:
8934:
8884:
8844:
8804:
8772:
8740:
8713:
8682:
8655:
8618:
8584:
8536:
8504:
8477:
8354:
8327:
8301:
8206:
8205:{\textstyle X/\sigma }
8175:
8114:
8063:
8037:
7968:
7954:The absolute value of
7945:
7870:
7843:
7772:
7753:linear transformations
7745:
7708:
7679:
7659:
7639:
7608:
7601:
7580:
7543:
7507:
7469:
7431:
7398:
7362:
7342:
7318:
7282:
7249:
7216:
7183:
7150:
7130:
7110:
7079:
7078:{\textstyle \sigma =3}
7053:
7024:
7004:
6955:
6935:
6900:
6880:
6857:
6837:
6795:
6781:, for large values of
6775:
6755:
6735:
6708:
6688:
6668:
6630:
6603:
6561:
6554:
6531:
6511:
6481:
6455:
6387:
6345:
6296:
6263:
6226:
6171:
5934:
5914:
5854:
5809:
5757:
5639:
5612:
5586:
5345:
5113:
5046:
4968:
4948:
4916:
4896:
4869:
4849:
4817:
4788:
4731:
4711:
4710:{\textstyle \phi _{X}}
4647:
4616:
4489:
4468:
4364:
4333:
4232:
4211:
4133:
4102:
4027:
4006:
3954:
3926:
3880:
3859:
3808:
3762:
3735:
3715:
3684:
3645:
3538:
3465:
3385:
3350:
3318:
3292:
3187:
3104:
3082:
3053:
3024:
2972:
2675:
2647:
2623:
2595:
2171:
2125:
2098:
2029:
2010:
1980:
1944:
1917:
1888:
1637:Continuous or discrete
1590:Bernoulli distribution
1450:
1314:
1121:
989:
912:
838:
772:
744:
716:
673:
638:
610:
582:
554:
479:
355:
252:
206:
152:
116:
19686:Bivariate (spherical)
19184:Kaniadakis Îș-Gaussian
18458:"Normal distribution"
18408:on February 29, 2012.
18390:West, Graeme (2009).
18379:"Normal Distribution"
18359:10.1145/225545.225554
18129:10.1081/sta-200052102
18078:10.3917/pope.303.0303
17871:10.18637/jss.v011.i04
17844:10.18637/jss.v005.i08
17729:10.1145/138351.138364
17576:10.1145/355744.355750
17407:"Normal Distribution"
17340:The Mismeasure of Man
17187:Dia, Yaya D. (2023).
17134:Statistical Inference
15689:Asymptotic Statistics
15484:"Normal Distribution"
15175:Standard normal table
15147:
15021:
14870:
14765:
14755:central limit theorem
14749:
14734:
14712:
14686:
14242:
14216:
14168:
14138:
14109:Chebyshev polynomials
14080:
13798:
13413:are returned. Again,
13408:
13242:
13093:central limit theorem
13043:
13031:Computational methods
13011:binomial distribution
12999:central limit theorem
12953:
12922:Nassim Nicholas Taleb
12842:
12830:
12781:Approximate normality
12759:
12730:
12588:
12564:central limit theorem
12496:sufficient statistics
12447:
12383:
12243:
12189:Sum of two quadratics
12120:AndersonâDarling test
12087:
11841:Statistical inference
11815:Kaniadakis statistics
11794:, and is one type of
11525:
11449:
11228:
11169:
11108:
10844:
10798:
10771:
10751:
10690:
10613:
10589:
10516:Rayleigh distribution
10509:
10460:Their Euclidean norm
10452:
10379:
10297:
10267:
10240:
10155:
10109:
10039:for a visualization).
10030:
9906:
9746:
9688:
9541:
9503:
9429:
9399:
9372:
9352:
9235:
9215:
9195:
9168:
9148:
9121:
9094:polarization identity
9087:
9057:
9031:
9005:
8978:
8958:
8935:
8885:
8845:
8805:
8773:
8741:
8739:{\textstyle \mu _{2}}
8714:
8712:{\textstyle \mu _{1}}
8683:
8656:
8619:
8585:
8537:
8505:
8478:
8355:
8328:
8302:
8207:
8176:
8115:
8069:this is known as the
8064:
8038:
7969:
7946:
7871:
7844:
7773:
7746:
7709:
7680:
7660:
7640:
7602:
7560:
7544:
7508:
7470:
7432:
7399:
7363:
7343:
7319:
7283:
7250:
7217:
7184:
7151:
7131:
7111:
7080:
7054:
7025:
7010:of a normal variable
7005:
6973:
6956:
6936:
6901:
6881:
6858:
6838:
6796:
6794:{\textstyle \lambda }
6776:
6774:{\textstyle \lambda }
6756:
6754:{\textstyle \lambda }
6736:
6734:{\textstyle \lambda }
6709:
6689:
6669:
6631:
6604:
6573:binomial distribution
6566:Central limit theorem
6555:
6532:
6512:
6487:
6479:
6472:Central limit theorem
6467:Related distributions
6456:
6388:
6346:
6297:
6264:
6232:. The same family is
6227:
6187:(NEF) with quadratic
6172:
5935:
5915:
5855:
5810:
5758:
5640:
5613:
5587:
5346:
5114:
5047:
4969:
4949:
4917:
4897:
4870:
4850:
4829:Marcinkiewicz theorem
4818:
4789:
4732:
4712:
4648:
4617:
4490:
4469:
4365:
4334:
4233:
4212:
4134:
4103:
4028:
4007:
3955:
3927:
3881:
3860:
3809:
3763:
3736:
3716:
3685:
3646:
3539:
3466:
3386:
3351:
3319:
3293:
3188:
3105:
3083:
3054:
3025:
2973:
2676:
2648:
2624:
2596:
2573:Further information:
2172:
2126:
2099:
2030:
2007:
1994:Further information:
1981:
1945:
1918:
1916:{\textstyle \Phi (x)}
1889:
1595:Binomial distribution
1451:
1315:
1122:
990:
913:
839:
773:
745:
717:
674:
639:
611:
583:
555:
480:
356:
253:
207:
153:
117:
52:The red curve is the
18:Gaussian distribution
19898:Stable distributions
19751:Dirac delta function
19698:Bivariate (toroidal)
19655:Univariate von Mises
19526:Multivariate Laplace
19418:Shifted log-logistic
18767:Continuous Bernoulli
17880:Maxwell, James Clerk
17201:10.2139/ssrn.4487559
16951:v. 6, paragraph 327.
16579:Krishnamoorthy (2006
16567:Krishnamoorthy (2006
16555:Krishnamoorthy (2006
16502:Quine, M.P. (1993).
15195:Tweedie distribution
15030:
14879:
14847:
14751:Pierre-Simon Laplace
14736:Carl Friedrich Gauss
14695:
14641:24.12333774572479110
14614:11.61511226260603247
14575:16.88639562007936908
14548:24.14804072812762821
14504:12.72323261907760928
14477:18.61193318971775795
14438:10.27157061171363079
14411:18.25323235347346525
14340:18.38871225773938487
14250:
14240:{\textstyle x\geq 0}
14225:
14176:
14166:{\textstyle 2^{-53}}
14147:
14136:{\textstyle 1-\Phi }
14121:
13891:
13586:
13326:
13133:
12956:distribution fitting
12936:standardized testing
12918:stock market crashes
12837:Iris flower data set
12788:infinitely divisible
12739:
12629:
12611:Dirac delta function
12416:
12266:
12203:
12151:Both univariate and
12004:
11887:Confidence intervals
11827:Pearson distribution
11796:Tsallis distribution
11694:stochastic processes
11471:
11255:
11251:degrees of freedom:
11178:
11119:
10853:
10849:degrees of freedom:
10827:
10780:
10760:
10701:
10622:
10602:
10539:
10464:
10395:
10309:
10280:
10249:
10168:
10162:product distribution
10121:
10053:
9915:
9755:
9697:
9553:
9512:
9441:
9412:
9381:
9361:
9244:
9224:
9204:
9177:
9157:
9130:
9103:
9066:
9040:
9014:
8987:
8967:
8947:
8894:
8854:
8814:
8782:
8750:
8723:
8696:
8665:
8638:
8598:
8552:
8514:
8494:
8368:
8344:
8311:
8220:
8188:
8128:
8080:
8047:
7982:
7958:
7884:
7860:
7786:
7762:
7718:
7707:{\textstyle a\mu +b}
7689:
7669:
7649:
7620:
7557:
7517:
7479:
7441:
7408:
7372:
7352:
7332:
7292:
7259:
7226:
7193:
7160:
7140:
7120:
7093:
7063:
7052:{\textstyle \mu =-2}
7034:
7014:
6981:
6945:
6934:{\textstyle t(\nu )}
6916:
6890:
6867:
6847:
6811:
6785:
6765:
6745:
6725:
6719:Poisson distribution
6698:
6678:
6667:{\textstyle np(1-p)}
6640:
6617:
6611:approximately normal
6578:
6541:
6521:
6492:
6397:
6355:
6309:
6273:
6240:
6213:
6204:statistical manifold
6200:information geometry
5944:
5924:
5864:
5819:
5773:
5649:
5622:
5602:
5359:
5123:
5056:
4989:
4958:
4938:
4906:
4886:
4859:
4839:
4798:
4741:
4721:
4694:
4627:
4505:
4479:
4380:
4344:
4248:
4222:
4149:
4113:
4043:
4017:
3970:
3937:
3896:
3870:
3849:
3772:
3745:
3725:
3705:
3655:
3554:
3546:More generally, its
3477:
3417:
3360:
3328:
3308:
3300:Its density has two
3199:
3117:
3094:
3081:{\textstyle x=\mu .}
3063:
3034:
3005:
2990:of the distribution.
2971:{\textstyle x=\mu ,}
2953:
2658:
2637:
2606:
2585:
2136:
2109:
2040:
2019:
2000:Coverage probability
1954:
1927:
1898:
1834:
1721:Law of large numbers
1690:Marginal probability
1615:Poisson distribution
1464:Part of a series on
1332:
1139:
1007:
930:
856:
790:
762:
734:
691:
656:
637:{\displaystyle \mu }
628:
609:{\displaystyle \mu }
600:
581:{\displaystyle \mu }
572:
497:
373:
270:
234:
171:
134:
80:
19878:Normal distribution
19799:Natural exponential
19704:Bivariate von Mises
19670:Wrapped exponential
19536:Multivariate stable
19531:Multivariate normal
18852:Benktander 2nd kind
18847:Benktander 1st kind
18636:Discrete phase-type
18203:Stigler, Stephen M.
17807:. Springer-Verlag.
17645:Statistical Science
17327:English translation
17278:Natural Inheritance
17117:. Springer-Verlag.
16707:, Equation (26.48))
16478:, Theorem 3.5)
16332:Weisstein, Eric W.
16289:10.1167/jov.21.10.1
16086:O'Hagan, A. (1994)
15417:Normal Distribution
14710:{\textstyle x<0}
14632:4.92081346632882033
14605:3.83362947800146179
14566:5.26184239579604207
14539:4.91396098895240075
14495:5.51862483025707963
14468:5.66479518878470765
14429:5.70347935898051437
14402:7.30756258553673541
14367:8.97280659046817350
14358:5.81582518933527391
14331:8.42742300458043240
14301:2.92678600515804815
14291:0.39894228040143268
13532:ideal approximation
13263:) the squared norm
13091:that relies on the
12894:BlackâScholes model
12613:), then after time
12522:pseudo-observations
12479:With known variance
12167:prior distributions
11817:, being one of the
11711:. A random element
11497:
11405:
11381:
11363:
11328:
11304:
11286:
10681:
10663:
10639:
10618:degrees of freedom
10501:
10483:
10389:Cauchy distribution
10021:
9988:
9968:
9953:
9932:
9897:
9864:
9844:
9801:
9737:
9682:
9661:
9625:
9595:
9494:
8929:
8911:
8799:
8767:
8326:{\textstyle \mu =0}
8265:
8062:{\textstyle \mu =0}
7758:The exponential of
7496:
7458:
6602:{\textstyle B(n,p)}
6145:
6093:
6044:
5906:
5573:
5555:
5471:
5453:
5333:
5318:
5284:
5269:
5231:
5105:
5038:
4833:JĂłzef Marcinkiewicz
4674:Zero-variance limit
1996:Interval estimation
1680:Complementary event
1622:Probability measure
1610:Pareto distribution
1605:Normal distribution
1302:
1287:
1253:
39:
38:Normal distribution
19454:Rectified Gaussian
19339:Generalized Pareto
19197:Generalized normal
19069:Matrix-exponential
18376:Weisstein, Eric W.
17335:Gould, Stephen Jay
17217:de Moivre, Abraham
16928:Jaynes, Edwin J.;
16672:Applied Statistics
16160:. December 5, 2007
16123:UIUC, Lecture 21.
15625:Abramowitz, Milton
15601:Weisstein, Eric W.
15588:on March 25, 2009.
15488:www.mathsisfun.com
15319:, chapter V)) and
15142:
15129:
15016:
14865:
14799:Bates distribution
14791:Mathematics portal
14759:
14744:
14707:
14681:
14679:
14237:
14211:
14163:
14133:
14075:
13869:continued fraction
13793:
13576:| < 7.5·10
13539:Hadamard transform
13524:ziggurat algorithm
13493:+ 1.4 then reject
13403:
13237:
13054:
13015:plotting positions
12988:Bell curve grading
12959:
12928:Measurement errors
12912:, which possesses
12841:
12754:
12725:
12623:diffusion equation
12595:
12550:
12442:
12378:
12248:has the form of a
12238:
12082:
12081:
11690:Gaussian processes
11520:
11483:
11444:
11391:
11367:
11349:
11314:
11290:
11272:
11223:
11164:
11103:
10839:
10809:standard deviation
10793:
10766:
10746:
10685:
10667:
10649:
10625:
10608:
10584:
10530:linear combination
10504:
10487:
10469:
10447:
10374:
10292:
10265:{\textstyle K_{0}}
10262:
10235:
10150:
10104:
10025:
10007:
9974:
9954:
9939:
9918:
9901:
9883:
9850:
9830:
9787:
9741:
9723:
9683:
9662:
9647:
9605:
9581:
9536:
9498:
9480:
9424:
9394:
9367:
9347:
9230:
9210:
9190:
9163:
9146:{\textstyle X_{2}}
9143:
9119:{\textstyle X_{1}}
9116:
9082:
9052:
9026:
9000:
8973:
8953:
8943:In particular, if
8930:
8915:
8897:
8880:
8840:
8800:
8785:
8768:
8753:
8736:
8709:
8681:{\textstyle X_{2}}
8678:
8654:{\textstyle X_{1}}
8651:
8614:
8580:
8532:
8500:
8473:
8350:
8323:
8297:
8251:
8202:
8171:
8110:
8059:
8033:
7964:
7941:
7866:
7839:
7768:
7741:
7704:
7675:
7655:
7635:
7609:
7597:
7539:
7503:
7482:
7465:
7444:
7427:
7394:
7358:
7338:
7314:
7278:
7245:
7212:
7179:
7146:
7126:
7109:{\textstyle x^{y}}
7106:
7075:
7049:
7020:
7000:
6951:
6931:
6896:
6876:
6853:
6833:
6791:
6771:
6751:
6731:
6704:
6684:
6664:
6626:
6599:
6562:
6550:
6527:
6507:
6482:
6451:
6415:
6383:
6341:
6292:
6259:
6222:
6208:constant curvature
6181:exponential family
6167:
6131:
6079:
6030:
5930:
5910:
5892:
5850:
5805:
5753:
5747:
5635:
5608:
5582:
5559:
5541:
5457:
5439:
5353:Hellinger distance
5341:
5319:
5304:
5270:
5255:
5217:
5109:
5091:
5042:
5024:
4964:
4944:
4912:
4892:
4865:
4845:
4813:
4784:
4727:
4707:
4643:
4612:
4485:
4464:
4360:
4329:
4228:
4207:
4129:
4098:
4023:
4002:
3950:
3922:
3876:
3855:
3835:Non-central moment
3804:
3758:
3731:
3711:
3696:Hermite polynomial
3680:
3641:
3534:
3461:
3381:
3346:
3314:
3288:
3183:
3100:
3078:
3049:
3020:
2968:
2674:{\textstyle z_{p}}
2671:
2643:
2622:{\textstyle z_{p}}
2619:
2591:
2167:
2121:
2094:
2025:
2011:
1976:
1943:{\textstyle x_{0}}
1940:
1913:
1884:
1731:Boole's inequality
1667:Stochastic process
1556:Mutual exclusivity
1473:Probability theory
1446:
1325:Expected shortfall
1310:
1288:
1273:
1239:
1117:
1111:
1000:Fisher information
985:
908:
834:
768:
740:
712:
669:
634:
606:
578:
550:
475:
351:
248:
202:
148:
112:
37:
19865:
19864:
19462:
19461:
19431:
19430:
19322:whose type varies
19268:Normal (Gaussian)
19222:Hyperbolic secant
19171:Exponential power
19074:MaxwellâBoltzmann
18822:Wigner semicircle
18714:
18713:
18686:Parabolic fractal
18676:Negative binomial
18440:978-0-486-61272-0
18329:978-0-486-64690-9
18303:978-0-674-83601-3
18284:978-0-674-40340-6
18187:10.1002/wics.1199
18175:WIREs Comput Stat
18146:WIREs Comput Stat
17941:978-0-8247-9342-5
17823:Marsaglia, George
17814:978-0-387-97137-7
17759:. Paris: 447â462.
17746:on July 16, 2010.
17690:978-0-387-95036-5
17622:978-0-19-852341-3
17603:978-1-58488-635-8
17510:978-0-471-58494-0
17491:978-0-471-58495-7
17464:978-0-486-61114-3
17445:978-0-02-914673-6
17398:978-0-88275-642-4
17350:978-0-393-01489-1
17304:978-0-8247-5402-0
17232:978-0-8218-2103-9
17143:978-0-534-24312-8
17124:978-0-387-97990-8
17105:978-0-471-49464-5
17086:978-0-8218-0531-2
16647:978-90-70754-33-4
16450:978-0-387-94919-2
16267:Journal of Vision
16090:, Edward Arnold.
16059:978-0-471-49464-5
16026:978-0-521-00618-7
15707:978-0-511-80225-6
15648:978-0-486-61272-0
15629:Stegun, Irene Ann
15441:Lyon, A. (2014).
15263:in Seriem Expansi
15101:
15014:
15005:
15004:
14990:
14915:
14817:ErdĆsâKac theorem
14673:
14644:
14578:
14507:
14441:
14370:
14304:
14062:
14022:
13988:
13960:
13917:
13788:
13401:
13400:
13362:
13361:
13210:
13159:
13108:BoxâMuller method
12908:have argued that
12906:Benoit Mandelbrot
12902:compound interest
12896:, changes in the
12823:Assumed normality
12812:Thermal radiation
12757:{\textstyle g(x)}
12705:
12676:
12645:
12548:
12518:conditional prior
12440:
12325:
12322:
12309:
12290:
12236:
12108:ShapiroâWilk test
12078:
12058:
11750:operator K: H â H
11420:
11079:
11078:
11061:
11016:
10985:
10915:
10899:
10896:
10873:
10817:Cochran's theorem
10769:{\textstyle \mu }
10502:
10023:
9899:
9645:
9639:
9370:{\textstyle \mu }
9339:
9338:
9166:{\textstyle \mu }
8810:, then their sum
8592:LĂ©vy distribution
8463:
8426:
8403:
7638:{\textstyle aX+b}
6954:{\textstyle \nu }
6510:{\textstyle p(k)}
6449:
6406:
6189:variance function
6146:
6121:
6095:
6074:
6054:
6015:
5933:{\textstyle \mu }
5860:and the prior is
5743:
5707:
5611:{\textstyle \mu }
5575:
5498:
5474:
5473:
5334:
5285:
5246:
5233:
4816:{\textstyle Q(t)}
4656:
4655:
3858:{\textstyle \mu }
3734:{\textstyle \mu }
3694:th (probabilist)
3550:th derivative is
3302:inflection points
3271:
3166:
3059:and zero only at
2936:
2935:
2569:Quantile function
2566:
2565:
2562:
2561:
2505:
2504:
2445:
2444:
2377:
2376:
2309:
2308:
2241:
2240:
2150:
2142:
1783:
1782:
1685:Joint probability
1632:Bernoulli process
1531:Probability space
1459:
1458:
1444:
1429:
1408:
1360:
1359:
1303:
1254:
1184:
1150:
801:
771:{\displaystyle 0}
743:{\displaystyle 0}
710:
514:
464:
461:
416:
399:
347:
296:
295:
16:(Redirected from
19910:
19855:
19854:
19845:
19844:
19784:Compound Poisson
19759:
19747:
19716:von MisesâFisher
19712:
19700:
19688:
19650:Circular uniform
19646:
19566:
19510:
19481:
19442:
19441:
19344:MarchenkoâPastur
19207:Geometric stable
19124:Truncated normal
19017:Inverse Gaussian
18923:Hyperexponential
18762:Beta rectangular
18730:bounded interval
18725:
18724:
18593:Discrete uniform
18578:Poisson binomial
18529:
18528:
18504:
18497:
18490:
18481:
18480:
18471:
18444:
18409:
18407:
18399:Wilmott Magazine
18396:
18386:
18371:
18361:
18333:
18317:
18307:
18288:
18276:
18265:
18236:
18226:
18198:
18169:
18158:10.1002/wics.151
18140:
18111:
18082:
18080:
18055:
18022:
17986:
17958:
17945:
17926:
17924:
17915:(172): 559â568.
17899:
17875:
17873:
17848:
17846:
17818:
17806:
17795:
17785:
17760:
17747:
17745:
17739:. Archived from
17722:
17704:
17694:
17675:
17642:
17626:
17607:
17588:
17578:
17553:
17535:
17514:
17495:
17476:
17449:
17420:
17402:
17383:
17354:
17330:
17308:
17296:
17285:
17283:
17271:
17261:
17252:(3): 1257â1272.
17236:
17212:
17183:
17174:
17172:
17163:(107): 631â638.
17147:
17128:
17109:
17090:
17054:
17045:
17025:
17024:
16999:(5): 1591â1613.
16984:
16978:
16977:
16970:
16964:
16958:
16952:
16948:Collected Papers
16943:
16937:
16926:
16920:
16914:
16908:
16902:
16896:
16890:
16884:
16878:
16872:
16866:
16860:
16854:
16848:
16842:
16836:
16830:
16824:
16818:
16812:
16805:
16799:
16793:
16787:
16781:
16775:
16770:
16764:
16758:
16752:
16747:
16741:
16736:
16730:
16725:
16719:
16714:
16708:
16702:
16696:
16695:
16667:
16661:
16658:
16652:
16651:
16635:
16626:
16620:
16619:
16599:
16593:
16588:
16582:
16576:
16570:
16564:
16558:
16552:
16543:
16542:
16522:
16516:
16515:
16499:
16493:
16488:
16479:
16473:
16467:
16461:
16455:
16454:
16436:
16430:
16429:
16399:
16393:
16392:
16374:
16348:
16342:
16341:
16329:
16323:
16317:
16311:
16310:
16300:
16282:
16258:
16249:
16248:
16246:
16244:
16230:
16224:
16223:
16221:
16219:
16205:
16199:
16194:
16188:
16187:
16185:
16176:
16170:
16169:
16167:
16165:
16150:
16144:
16135:
16129:
16120:
16114:
16108:
16099:
16084:
16078:
16077:
16071:
16063:
16047:
16037:
16031:
16030:
16014:
16004:
15998:
15992:
15986:
15980:
15974:
15968:
15962:
15961:
15959:
15947:
15941:
15940:
15932:
15926:
15920:
15914:
15908:
15902:
15896:
15885:
15884:
15819:
15813:
15810:
15804:
15803:
15801:
15799:
15794:on March 7, 2016
15793:
15787:. Archived from
15778:
15760:
15751:
15745:
15744:
15728:
15718:
15712:
15711:
15683:
15677:
15676:
15631:, eds. (1983) .
15621:
15615:
15614:
15613:
15596:
15590:
15589:
15587:
15580:
15571:
15565:
15564:
15557:"The Q-function"
15552:
15546:
15540:
15534:
15528:
15522:
15516:
15510:
15505:
15499:
15498:
15496:
15494:
15480:
15471:
15470:
15466:978-0387-30303-1
15452:
15446:
15439:
15433:
15427:
15421:
15413:
15407:
15406:
15404:
15402:
15381:
15363:
15354:
15332:
15305:
15299:
15288:
15282:
15275:
15261:
15248:
15241:
15235:
15228:
15151:
15149:
15148:
15143:
15141:
15137:
15130:
15107:
15103:
15102:
15094:
15072:
15071:
15062:
15061:
15056:
15025:
15023:
15022:
15017:
15015:
15013:
15012:
15011:
15007:
15006:
15000:
14996:
14991:
14983:
14971:
14958:
14957:
14933:
14932:
14917:
14916:
14908:
14898:
14874:
14872:
14871:
14866:
14843:with the pdf on
14793:
14788:
14787:
14772:
14716:
14714:
14713:
14708:
14690:
14688:
14687:
14682:
14680:
14676:
14675:
14674:
14669:
14668:
14659:
14649:
14645:
14643:
14627:
14626:
14616:
14600:
14599:
14589:
14583:
14579:
14577:
14561:
14560:
14550:
14534:
14533:
14523:
14516:
14512:
14508:
14506:
14490:
14489:
14479:
14463:
14462:
14452:
14446:
14442:
14440:
14424:
14423:
14413:
14397:
14396:
14386:
14379:
14375:
14371:
14369:
14353:
14352:
14342:
14326:
14325:
14315:
14309:
14305:
14303:
14289:
14276:
14246:
14244:
14243:
14238:
14220:
14218:
14217:
14212:
14210:
14206:
14205:
14204:
14172:
14170:
14169:
14164:
14162:
14161:
14142:
14140:
14139:
14134:
14099:
14092:
14085:for calculating
14084:
14082:
14081:
14076:
14074:
14070:
14063:
14061:
14038:
14037:
14028:
14023:
14021:
14004:
14003:
13994:
13989:
13987:
13976:
13975:
13966:
13961:
13956:
13955:
13946:
13918:
13910:
13885:Marsaglia (2004)
13862:
13846:= â1.821255978,
13832:= â0.356563782,
13802:
13800:
13799:
13794:
13789:
13787:
13783:
13782:
13763:
13736:
13732:
13731:
13730:
13721:
13720:
13708:
13707:
13698:
13697:
13685:
13684:
13675:
13674:
13662:
13661:
13652:
13651:
13636:
13635:
13577:
13575:
13451:
13450:
13412:
13410:
13409:
13404:
13402:
13396:
13379:
13378:
13363:
13357:
13340:
13339:
13318:is computed. If
13317:
13272:
13253:bivariate normal
13246:
13244:
13243:
13238:
13211:
13194:
13160:
13143:
12964:percentile ranks
12849:
12763:
12761:
12760:
12755:
12734:
12732:
12731:
12726:
12706:
12704:
12703:
12702:
12689:
12688:
12679:
12677:
12669:
12646:
12644:
12633:
12534:conjugate priors
12452:is one-half the
12451:
12449:
12448:
12443:
12441:
12439:
12428:
12420:
12387:
12385:
12384:
12379:
12374:
12373:
12361:
12360:
12345:
12344:
12326:
12324:
12323:
12315:
12310:
12302:
12296:
12291:
12289:
12278:
12270:
12250:weighted average
12247:
12245:
12244:
12239:
12237:
12235:
12224:
12207:
12103:JarqueâBera test
12091:
12089:
12088:
12083:
12080:
12079:
12071:
12068:
12060:
12059:
12051:
12045:
12044:
12023:
12022:
11916:, also known as
11751:
11745:
11730:
11720:
11710:
11659:
11605:
11594:
11593:
11580:
11529:
11527:
11526:
11521:
11513:
11512:
11496:
11491:
11453:
11451:
11450:
11445:
11440:
11439:
11421:
11419:
11415:
11410:
11406:
11404:
11399:
11380:
11375:
11362:
11357:
11342:
11338:
11333:
11329:
11327:
11322:
11303:
11298:
11285:
11280:
11265:
11250:
11238:
11232:
11230:
11229:
11224:
11222:
11221:
11203:
11202:
11190:
11189:
11173:
11171:
11170:
11165:
11163:
11162:
11144:
11143:
11131:
11130:
11112:
11110:
11109:
11104:
11099:
11098:
11080:
11077:
11073:
11072:
11071:
11062:
11054:
11049:
11048:
11027:
11026:
11017:
11009:
11004:
11003:
10986:
10984:
10961:
10959:
10958:
10948:
10947:
10929:
10928:
10916:
10908:
10905:
10900:
10898:
10897:
10892:
10890:
10881:
10874:
10866:
10863:
10848:
10846:
10845:
10842:{\textstyle n-1}
10840:
10802:
10800:
10799:
10794:
10792:
10791:
10775:
10773:
10772:
10767:
10755:
10753:
10752:
10747:
10745:
10744:
10726:
10725:
10713:
10712:
10694:
10692:
10691:
10686:
10680:
10675:
10662:
10657:
10638:
10633:
10617:
10615:
10614:
10609:
10593:
10591:
10590:
10585:
10583:
10582:
10564:
10563:
10551:
10550:
10513:
10511:
10510:
10505:
10503:
10500:
10495:
10482:
10477:
10468:
10456:
10454:
10453:
10448:
10422:
10421:
10412:
10407:
10406:
10383:
10381:
10380:
10375:
10373:
10372:
10368:
10352:
10351:
10321:
10320:
10301:
10299:
10298:
10295:{\textstyle z=0}
10293:
10271:
10269:
10268:
10263:
10261:
10260:
10244:
10242:
10241:
10236:
10231:
10223:
10215:
10214:
10205:
10204:
10180:
10179:
10159:
10157:
10156:
10151:
10149:
10148:
10139:
10138:
10113:
10111:
10110:
10105:
10088:
10087:
10078:
10077:
10065:
10064:
10034:
10032:
10031:
10026:
10024:
10022:
10020:
10015:
9987:
9982:
9969:
9967:
9962:
9952:
9947:
9937:
9931:
9926:
9910:
9908:
9907:
9902:
9900:
9898:
9896:
9891:
9863:
9858:
9845:
9843:
9838:
9829:
9828:
9800:
9795:
9786:
9785:
9772:
9767:
9766:
9750:
9748:
9747:
9742:
9736:
9731:
9719:
9718:
9706:
9705:
9692:
9690:
9689:
9684:
9681:
9670:
9660:
9655:
9646:
9644:
9640:
9637:
9624:
9613:
9594:
9589:
9580:
9579:
9578:
9577:
9572:
9557:
9545:
9543:
9542:
9537:
9507:
9505:
9504:
9499:
9493:
9488:
9476:
9475:
9463:
9462:
9453:
9452:
9433:
9431:
9430:
9425:
9403:
9401:
9400:
9395:
9393:
9392:
9376:
9374:
9373:
9368:
9356:
9354:
9353:
9348:
9340:
9337:
9336:
9324:
9323:
9314:
9313:
9291:
9290:
9275:
9274:
9261:
9256:
9255:
9239:
9237:
9236:
9231:
9219:
9217:
9216:
9211:
9199:
9197:
9196:
9191:
9189:
9188:
9172:
9170:
9169:
9164:
9152:
9150:
9149:
9144:
9142:
9141:
9125:
9123:
9122:
9117:
9115:
9114:
9091:
9089:
9088:
9083:
9081:
9080:
9061:
9059:
9058:
9055:{\textstyle X-Y}
9053:
9035:
9033:
9032:
9029:{\textstyle X+Y}
9027:
9009:
9007:
9006:
9001:
8999:
8998:
8982:
8980:
8979:
8974:
8962:
8960:
8959:
8954:
8939:
8937:
8936:
8931:
8928:
8923:
8910:
8905:
8889:
8887:
8886:
8881:
8879:
8878:
8866:
8865:
8849:
8847:
8846:
8841:
8839:
8838:
8826:
8825:
8809:
8807:
8806:
8801:
8798:
8793:
8777:
8775:
8774:
8769:
8766:
8761:
8745:
8743:
8742:
8737:
8735:
8734:
8718:
8716:
8715:
8710:
8708:
8707:
8687:
8685:
8684:
8679:
8677:
8676:
8660:
8658:
8657:
8652:
8650:
8649:
8623:
8621:
8620:
8615:
8613:
8612:
8589:
8587:
8586:
8581:
8579:
8578:
8541:
8539:
8538:
8533:
8509:
8507:
8506:
8501:
8482:
8480:
8479:
8474:
8469:
8465:
8464:
8456:
8437:
8436:
8431:
8427:
8422:
8411:
8404:
8396:
8359:
8357:
8356:
8351:
8332:
8330:
8329:
8324:
8306:
8304:
8303:
8298:
8293:
8292:
8283:
8278:
8277:
8264:
8259:
8247:
8246:
8237:
8232:
8231:
8211:
8209:
8208:
8203:
8198:
8180:
8178:
8177:
8172:
8170:
8169:
8154:
8149:
8135:
8122:chi distribution
8119:
8117:
8116:
8111:
8106:
8101:
8087:
8068:
8066:
8065:
8060:
8042:
8040:
8039:
8034:
8032:
8028:
8027:
8009:
8008:
7996:
7973:
7971:
7970:
7965:
7950:
7948:
7947:
7942:
7934:
7933:
7914:
7913:
7875:
7873:
7872:
7867:
7848:
7846:
7845:
7840:
7832:
7831:
7798:
7797:
7777:
7775:
7774:
7769:
7750:
7748:
7747:
7742:
7740:
7739:
7730:
7729:
7713:
7711:
7710:
7705:
7684:
7682:
7681:
7676:
7664:
7662:
7661:
7656:
7644:
7642:
7641:
7636:
7606:
7604:
7603:
7598:
7593:
7592:
7579:
7574:
7548:
7546:
7545:
7540:
7532:
7531:
7512:
7510:
7509:
7504:
7495:
7490:
7474:
7472:
7471:
7466:
7457:
7452:
7436:
7434:
7433:
7428:
7420:
7419:
7403:
7401:
7400:
7395:
7384:
7383:
7367:
7365:
7364:
7359:
7347:
7345:
7344:
7339:
7323:
7321:
7320:
7315:
7307:
7306:
7287:
7285:
7284:
7279:
7271:
7270:
7254:
7252:
7251:
7246:
7238:
7237:
7221:
7219:
7218:
7213:
7205:
7204:
7188:
7186:
7185:
7180:
7172:
7171:
7155:
7153:
7152:
7147:
7135:
7133:
7132:
7127:
7115:
7113:
7112:
7107:
7105:
7104:
7084:
7082:
7081:
7076:
7058:
7056:
7055:
7050:
7029:
7027:
7026:
7021:
7009:
7007:
7006:
7001:
6999:
6998:
6960:
6958:
6957:
6952:
6940:
6938:
6937:
6932:
6905:
6903:
6902:
6897:
6885:
6883:
6882:
6877:
6862:
6860:
6859:
6854:
6842:
6840:
6839:
6834:
6823:
6822:
6800:
6798:
6797:
6792:
6780:
6778:
6777:
6772:
6760:
6758:
6757:
6752:
6740:
6738:
6737:
6732:
6713:
6711:
6710:
6705:
6693:
6691:
6690:
6685:
6673:
6671:
6670:
6665:
6635:
6633:
6632:
6627:
6608:
6606:
6605:
6600:
6559:
6557:
6556:
6551:
6536:
6534:
6533:
6528:
6516:
6514:
6513:
6508:
6460:
6458:
6457:
6452:
6450:
6436:
6425:
6424:
6414:
6392:
6390:
6389:
6384:
6379:
6378:
6350:
6348:
6347:
6342:
6340:
6339:
6321:
6320:
6301:
6299:
6298:
6293:
6291:
6290:
6268:
6266:
6265:
6260:
6258:
6257:
6231:
6229:
6228:
6223:
6176:
6174:
6173:
6168:
6166:
6162:
6161:
6160:
6152:
6148:
6147:
6144:
6139:
6127:
6122:
6120:
6119:
6107:
6096:
6094:
6092:
6087:
6075:
6070:
6069:
6060:
6057:
6056:
6055:
6047:
6043:
6038:
6026:
6025:
6016:
6011:
6010:
6001:
5998:
5991:
5990:
5981:
5980:
5962:
5961:
5939:
5937:
5936:
5931:
5919:
5917:
5916:
5911:
5905:
5900:
5888:
5887:
5859:
5857:
5856:
5851:
5846:
5845:
5814:
5812:
5811:
5806:
5804:
5803:
5785:
5784:
5762:
5760:
5759:
5754:
5752:
5751:
5744:
5742:
5741:
5740:
5724:
5708:
5706:
5705:
5693:
5677:
5676:
5658:
5657:
5644:
5642:
5641:
5636:
5634:
5633:
5617:
5615:
5614:
5609:
5591:
5589:
5588:
5583:
5581:
5577:
5576:
5574:
5572:
5567:
5554:
5549:
5539:
5538:
5537:
5528:
5527:
5515:
5514:
5501:
5499:
5491:
5475:
5472:
5470:
5465:
5452:
5447:
5437:
5436:
5435:
5426:
5425:
5412:
5411:
5397:
5396:
5384:
5383:
5371:
5370:
5350:
5348:
5347:
5342:
5340:
5336:
5335:
5332:
5327:
5317:
5312:
5303:
5286:
5283:
5278:
5268:
5263:
5254:
5247:
5239:
5234:
5232:
5230:
5225:
5212:
5211:
5210:
5201:
5200:
5188:
5187:
5174:
5166:
5165:
5153:
5152:
5140:
5139:
5138:
5118:
5116:
5115:
5110:
5104:
5099:
5087:
5086:
5068:
5067:
5051:
5049:
5048:
5043:
5037:
5032:
5020:
5019:
5001:
5000:
4973:
4971:
4970:
4965:
4953:
4951:
4950:
4945:
4930:, then they are
4921:
4919:
4918:
4913:
4901:
4899:
4898:
4893:
4874:
4872:
4871:
4866:
4854:
4852:
4851:
4846:
4822:
4820:
4819:
4814:
4793:
4791:
4790:
4785:
4753:
4752:
4736:
4734:
4733:
4728:
4716:
4714:
4713:
4708:
4706:
4705:
4684:Other properties
4652:
4650:
4649:
4644:
4642:
4641:
4621:
4619:
4618:
4613:
4611:
4610:
4595:
4594:
4585:
4584:
4569:
4568:
4559:
4558:
4543:
4542:
4533:
4532:
4517:
4516:
4494:
4492:
4491:
4486:
4473:
4471:
4470:
4465:
4463:
4462:
4444:
4443:
4434:
4433:
4418:
4417:
4408:
4407:
4392:
4391:
4369:
4367:
4366:
4361:
4359:
4358:
4338:
4336:
4335:
4330:
4328:
4327:
4312:
4311:
4302:
4301:
4286:
4285:
4276:
4275:
4260:
4259:
4237:
4235:
4234:
4229:
4216:
4214:
4213:
4208:
4206:
4205:
4187:
4186:
4177:
4176:
4161:
4160:
4138:
4136:
4135:
4130:
4128:
4127:
4107:
4105:
4104:
4099:
4097:
4096:
4081:
4080:
4071:
4070:
4055:
4054:
4032:
4030:
4029:
4024:
4011:
4009:
4008:
4003:
4001:
4000:
3982:
3981:
3959:
3957:
3956:
3951:
3949:
3948:
3931:
3929:
3928:
3923:
3921:
3920:
3908:
3907:
3885:
3883:
3882:
3877:
3864:
3862:
3861:
3856:
3829:
3828:
3813:
3811:
3810:
3805:
3800:
3767:
3765:
3764:
3759:
3757:
3756:
3740:
3738:
3737:
3732:
3720:
3718:
3717:
3712:
3693:
3689:
3687:
3686:
3681:
3667:
3666:
3650:
3648:
3647:
3642:
3613:
3612:
3603:
3602:
3572:
3571:
3549:
3543:
3541:
3540:
3535:
3512:
3511:
3487:
3470:
3468:
3467:
3462:
3427:
3390:
3388:
3387:
3382:
3355:
3353:
3352:
3347:
3323:
3321:
3320:
3315:
3297:
3295:
3294:
3289:
3272:
3270:
3269:
3260:
3259:
3258:
3246:
3245:
3223:
3209:
3192:
3190:
3189:
3184:
3167:
3165:
3164:
3155:
3144:
3127:
3109:
3107:
3106:
3101:
3087:
3085:
3084:
3079:
3058:
3056:
3055:
3050:
3029:
3027:
3026:
3021:
3001:is positive for
2977:
2975:
2974:
2969:
2932:
2931:
2928:
2925:
2915:
2914:
2911:
2908:
2896:
2895:
2892:
2889:
2879:
2878:
2875:
2872:
2860:
2859:
2856:
2853:
2843:
2842:
2839:
2836:
2824:
2823:
2820:
2817:
2807:
2806:
2803:
2800:
2788:
2787:
2784:
2781:
2771:
2770:
2767:
2764:
2752:
2751:
2748:
2745:
2735:
2734:
2731:
2728:
2716:
2715:
2712:
2709:
2699:
2698:
2695:
2692:
2680:
2678:
2677:
2672:
2670:
2669:
2652:
2650:
2649:
2644:
2628:
2626:
2625:
2620:
2618:
2617:
2600:
2598:
2597:
2592:
2579:
2578:
2558:
2553:
2552:
2549:
2542:
2538:
2537:
2534:
2531:
2524:
2523:
2520:
2517:
2501:
2500:
2493:
2492:
2489:
2482:
2478:
2477:
2474:
2471:
2464:
2463:
2460:
2457:
2441:
2440:
2433:
2432:
2425:
2421:
2420:
2417:
2414:
2407:
2406:
2403:
2400:
2388:
2373:
2372:
2369:
2362:
2357:
2353:
2352:
2349:
2346:
2339:
2338:
2335:
2332:
2320:
2305:
2304:
2301:
2294:
2289:
2285:
2284:
2281:
2278:
2271:
2270:
2267:
2264:
2252:
2237:
2236:
2233:
2230:
2223:
2218:
2214:
2213:
2210:
2207:
2200:
2199:
2196:
2193:
2176:
2174:
2173:
2168:
2151:
2148:
2143:
2140:
2130:
2128:
2127:
2124:{\textstyle 1-p}
2122:
2103:
2101:
2100:
2095:
2034:
2032:
2031:
2026:
2013:
2012:
1985:
1983:
1982:
1977:
1972:
1971:
1949:
1947:
1946:
1941:
1939:
1938:
1922:
1920:
1919:
1914:
1893:
1891:
1890:
1885:
1877:
1876:
1858:
1857:
1775:
1768:
1761:
1551:Elementary event
1483:
1461:
1460:
1455:
1453:
1452:
1447:
1445:
1443:
1432:
1431:
1430:
1425:
1424:
1423:
1418:
1414:
1413:
1409:
1404:
1393:
1387:
1386:
1367:
1361:
1352:
1348:
1345:
1319:
1317:
1316:
1311:
1309:
1305:
1304:
1301:
1296:
1286:
1281:
1272:
1255:
1252:
1247:
1238:
1237:
1236:
1227:
1226:
1214:
1213:
1200:
1195:
1194:
1189:
1185:
1183:
1182:
1173:
1172:
1163:
1151:
1143:
1126:
1124:
1123:
1118:
1116:
1115:
1105:
1104:
1089:
1067:
1066:
1057:
1035:
1034:
1016:
1015:
994:
992:
991:
986:
978:
973:
972:
963:
962:
917:
915:
914:
909:
901:
896:
895:
886:
885:
843:
841:
840:
835:
830:
829:
802:
794:
777:
775:
774:
769:
749:
747:
746:
741:
721:
719:
718:
713:
711:
706:
698:
678:
676:
675:
670:
668:
667:
643:
641:
640:
635:
615:
613:
612:
607:
587:
585:
584:
579:
559:
557:
556:
551:
528:
527:
515:
510:
484:
482:
481:
476:
474:
470:
469:
465:
463:
462:
457:
451:
440:
417:
409:
404:
400:
395:
384:
360:
358:
357:
352:
350:
349:
348:
346:
345:
344:
331:
330:
329:
307:
297:
294:
293:
278:
274:
257:
255:
254:
249:
247:
211:
209:
208:
203:
201:
200:
192:
183:
182:
157:
155:
154:
149:
147:
121:
119:
118:
113:
108:
107:
89:
88:
69:
50:
40:
36:
21:
19918:
19917:
19913:
19912:
19911:
19909:
19908:
19907:
19868:
19867:
19866:
19861:
19833:
19809:Maximum entropy
19767:
19755:
19743:
19733:
19725:
19708:
19696:
19684:
19639:
19626:
19563:Matrix-valued:
19560:
19506:
19477:
19469:
19458:
19446:
19437:
19427:
19321:
19315:
19232:
19158:
19156:
19150:
19079:MaxwellâJĂŒttner
18928:Hypoexponential
18834:
18832:
18831:supported on a
18826:
18787:Noncentral beta
18747:BaldingâNichols
18729:
18728:supported on a
18720:
18710:
18613:
18607:
18603:ZipfâMandelbrot
18533:
18524:
18518:
18508:
18456:
18453:
18448:
18447:
18441:
18405:
18394:
18330:
18315:
18304:
18285:
18254:10.2307/2684031
18100:10.2307/2347972
18011:10.2307/2331536
17969:(11): 559â572.
17956:
17942:
17815:
17743:
17720:10.1.1.544.5806
17702:
17691:
17623:
17604:
17542:10.1145/2710016
17511:
17492:
17465:
17446:
17426:Murray, Charles
17405:
17399:
17372:10.2307/2681417
17351:
17305:
17281:
17233:
17144:
17125:
17106:
17096:Bayesian Theory
17087:
17034:
17029:
17028:
16985:
16981:
16972:
16971:
16967:
16959:
16955:
16944:
16940:
16927:
16923:
16915:
16911:
16903:
16899:
16891:
16887:
16879:
16875:
16867:
16863:
16855:
16851:
16843:
16839:
16831:
16827:
16819:
16815:
16806:
16802:
16794:
16790:
16782:
16778:
16771:
16767:
16759:
16755:
16748:
16744:
16737:
16733:
16726:
16722:
16715:
16711:
16703:
16699:
16684:10.2307/2347330
16668:
16664:
16659:
16655:
16648:
16633:
16627:
16623:
16616:
16600:
16596:
16589:
16585:
16577:
16573:
16565:
16561:
16553:
16546:
16523:
16519:
16500:
16496:
16489:
16482:
16474:
16470:
16462:
16458:
16451:
16437:
16433:
16400:
16396:
16349:
16345:
16330:
16326:
16318:
16314:
16259:
16252:
16242:
16240:
16232:
16231:
16227:
16217:
16215:
16207:
16206:
16202:
16195:
16191:
16183:
16177:
16173:
16163:
16161:
16152:
16151:
16147:
16136:
16132:
16121:
16117:
16109:
16102:
16085:
16081:
16065:
16064:
16060:
16044:Bayesian theory
16038:
16034:
16027:
16005:
16001:
15993:
15989:
15981:
15977:
15969:
15965:
15948:
15944:
15933:
15929:
15921:
15917:
15913:, p. 1258)
15909:
15905:
15897:
15888:
15820:
15816:
15811:
15807:
15797:
15795:
15791:
15776:10.1.1.511.9750
15758:
15752:
15748:
15741:
15719:
15715:
15708:
15684:
15680:
15649:
15622:
15618:
15597:
15593:
15585:
15578:
15572:
15568:
15553:
15549:
15541:
15537:
15531:McPherson (1990
15529:
15525:
15517:
15513:
15506:
15502:
15492:
15490:
15482:
15481:
15474:
15467:
15453:
15449:
15440:
15436:
15428:
15424:
15414:
15410:
15400:
15398:
15361:
15355:
15351:
15346:
15341:
15336:
15335:
15306:
15302:
15298:, p. 189)
15289:
15285:
15281:, section 177)
15276:
15272:
15246:
15242:
15238:
15229:
15225:
15220:
15128:
15127:
15109:
15108:
15093:
15086:
15082:
15078:
15077:
15073:
15067:
15063:
15057:
15055:
15054:
15031:
15028:
15027:
14995:
14982:
14981:
14977:
14976:
14972:
14953:
14949:
14922:
14918:
14907:
14903:
14899:
14897:
14880:
14877:
14876:
14848:
14845:
14844:
14789:
14782:
14779:
14773:
14768:
14764:
14729:
14724:
14696:
14693:
14692:
14678:
14677:
14664:
14660:
14658:
14654:
14650:
14622:
14618:
14617:
14595:
14591:
14590:
14588:
14584:
14556:
14552:
14551:
14529:
14525:
14524:
14522:
14518:
14514:
14513:
14485:
14481:
14480:
14458:
14454:
14453:
14451:
14447:
14419:
14415:
14414:
14392:
14388:
14387:
14385:
14381:
14377:
14376:
14348:
14344:
14343:
14321:
14317:
14316:
14314:
14310:
14293:
14288:
14284:
14277:
14266:
14253:
14251:
14248:
14247:
14226:
14223:
14222:
14197:
14193:
14183:
14179:
14177:
14174:
14173:
14154:
14150:
14148:
14145:
14144:
14122:
14119:
14118:
14094:
14086:
14039:
14033:
14029:
14027:
14005:
13999:
13995:
13993:
13977:
13971:
13967:
13965:
13951:
13947:
13945:
13938:
13934:
13909:
13892:
13889:
13888:
13860:
13852:
13845:
13839:= 1.781477937,
13838:
13831:
13825:= 0.319381530,
13824:
13817:
13778:
13774:
13767:
13762:
13726:
13722:
13716:
13712:
13703:
13699:
13693:
13689:
13680:
13676:
13670:
13666:
13657:
13653:
13647:
13643:
13631:
13627:
13626:
13622:
13587:
13584:
13583:
13566:
13564:
13549:
13445:
13443:
13380:
13377:
13341:
13338:
13327:
13324:
13323:
13305:
13264:
13255:random vector (
13193:
13142:
13134:
13131:
13130:
13071:probit function
13038:
13033:
13028:
13017:as part of the
13007:confidence belt
12850:
12845:
12833:Iris versicolor
12825:
12783:
12740:
12737:
12736:
12698:
12694:
12690:
12684:
12680:
12678:
12668:
12637:
12632:
12630:
12627:
12626:
12583:
12581:Exact normality
12571:maximum entropy
12556:
12551:
12491:
12486:
12484:With known mean
12481:
12476:
12471:
12429:
12421:
12419:
12417:
12414:
12413:
12366:
12362:
12353:
12349:
12337:
12333:
12314:
12301:
12300:
12295:
12279:
12271:
12269:
12267:
12264:
12263:
12225:
12208:
12206:
12204:
12201:
12200:
12196:
12191:
12138:
12125:Lilliefors test
12070:
12069:
12064:
12050:
12049:
12034:
12030:
12012:
12008:
12005:
12002:
12001:
11998:
11992:
11954:
11947:
11941:
11929:
11910:
11908:Normality tests
11904:
11902:Normality tests
11899:
11889:
11884:
11874:
11872:Sample variance
11869:
11863:
11858:
11848:
11843:
11792:Tsallis entropy
11763:Brownian bridge
11757:Brownian motion
11749:
11735:
11722:
11712:
11705:
11651:
11603:
11599:
11592:
11586:
11585:
11584:
11582:
11572:
11569:Euclidean space
11556:
11551:
11546:
11541:
11508:
11504:
11492:
11487:
11472:
11469:
11468:
11460:
11429:
11425:
11411:
11400:
11395:
11376:
11371:
11358:
11353:
11348:
11344:
11343:
11334:
11323:
11318:
11299:
11294:
11281:
11276:
11271:
11267:
11266:
11264:
11256:
11253:
11252:
11240:
11234:
11217:
11213:
11198:
11194:
11185:
11181:
11179:
11176:
11175:
11158:
11154:
11139:
11135:
11126:
11122:
11120:
11117:
11116:
11088:
11084:
11067:
11063:
11053:
11044:
11040:
11022:
11018:
11008:
10999:
10995:
10991:
10987:
10965:
10960:
10943:
10939:
10924:
10920:
10907:
10906:
10904:
10891:
10886:
10882:
10865:
10864:
10862:
10854:
10851:
10850:
10828:
10825:
10824:
10787:
10783:
10781:
10778:
10777:
10761:
10758:
10757:
10740:
10736:
10721:
10717:
10708:
10704:
10702:
10699:
10698:
10676:
10671:
10658:
10653:
10634:
10629:
10623:
10620:
10619:
10603:
10600:
10599:
10578:
10574:
10559:
10555:
10546:
10542:
10540:
10537:
10536:
10525:
10496:
10491:
10478:
10473:
10467:
10465:
10462:
10461:
10417:
10413:
10408:
10402:
10398:
10396:
10393:
10392:
10364:
10357:
10353:
10347:
10343:
10316:
10312:
10310:
10307:
10306:
10281:
10278:
10277:
10256:
10252:
10250:
10247:
10246:
10227:
10219:
10210:
10206:
10197:
10193:
10175:
10171:
10169:
10166:
10165:
10144:
10140:
10134:
10130:
10122:
10119:
10118:
10083:
10082:
10073:
10069:
10060:
10056:
10054:
10051:
10050:
10046:
10016:
10011:
9983:
9978:
9970:
9963:
9958:
9948:
9943:
9938:
9936:
9927:
9922:
9916:
9913:
9912:
9892:
9887:
9859:
9854:
9846:
9839:
9834:
9824:
9820:
9796:
9791:
9781:
9777:
9773:
9771:
9762:
9758:
9756:
9753:
9752:
9732:
9727:
9714:
9710:
9701:
9700:
9698:
9695:
9694:
9671:
9666:
9656:
9651:
9636:
9614:
9609:
9590:
9585:
9573:
9568:
9567:
9566:
9562:
9561:
9556:
9554:
9551:
9550:
9513:
9510:
9509:
9489:
9484:
9471:
9467:
9458:
9457:
9448:
9444:
9442:
9439:
9438:
9413:
9410:
9409:
9408:(with exponent
9388:
9384:
9382:
9379:
9378:
9362:
9359:
9358:
9332:
9328:
9319:
9315:
9286:
9282:
9270:
9266:
9262:
9260:
9251:
9247:
9245:
9242:
9241:
9225:
9222:
9221:
9205:
9202:
9201:
9184:
9180:
9178:
9175:
9174:
9158:
9155:
9154:
9137:
9133:
9131:
9128:
9127:
9110:
9106:
9104:
9101:
9100:
9076:
9072:
9067:
9064:
9063:
9041:
9038:
9037:
9015:
9012:
9011:
8994:
8990:
8988:
8985:
8984:
8968:
8965:
8964:
8948:
8945:
8944:
8924:
8919:
8906:
8901:
8895:
8892:
8891:
8874:
8870:
8861:
8857:
8855:
8852:
8851:
8834:
8830:
8821:
8817:
8815:
8812:
8811:
8794:
8789:
8783:
8780:
8779:
8762:
8757:
8751:
8748:
8747:
8730:
8726:
8724:
8721:
8720:
8703:
8699:
8697:
8694:
8693:
8672:
8668:
8666:
8663:
8662:
8645:
8641:
8639:
8636:
8635:
8631:
8605:
8601:
8599:
8596:
8595:
8571:
8567:
8553:
8550:
8549:
8515:
8512:
8511:
8495:
8492:
8491:
8455:
8451:
8447:
8432:
8412:
8410:
8406:
8405:
8395:
8369:
8366:
8365:
8345:
8342:
8341:
8312:
8309:
8308:
8288:
8284:
8279:
8273:
8269:
8260:
8255:
8242:
8238:
8233:
8227:
8223:
8221:
8218:
8217:
8194:
8189:
8186:
8185:
8165:
8161:
8150:
8145:
8131:
8129:
8126:
8125:
8102:
8097:
8083:
8081:
8078:
8077:
8048:
8045:
8044:
8023:
8019:
8004:
8000:
7986:
7985:
7983:
7980:
7979:
7959:
7956:
7955:
7929:
7925:
7909:
7908:
7885:
7882:
7881:
7861:
7858:
7857:
7827:
7823:
7793:
7789:
7787:
7784:
7783:
7778:is distributed
7763:
7760:
7759:
7735:
7731:
7725:
7721:
7719:
7716:
7715:
7690:
7687:
7686:
7670:
7667:
7666:
7650:
7647:
7646:
7621:
7618:
7617:
7614:
7588:
7584:
7575:
7564:
7558:
7555:
7554:
7524:
7520:
7518:
7515:
7514:
7491:
7486:
7480:
7477:
7476:
7453:
7448:
7442:
7439:
7438:
7415:
7411:
7409:
7406:
7405:
7379:
7375:
7373:
7370:
7369:
7353:
7350:
7349:
7333:
7330:
7329:
7299:
7295:
7293:
7290:
7289:
7266:
7262:
7260:
7257:
7256:
7233:
7229:
7227:
7224:
7223:
7200:
7196:
7194:
7191:
7190:
7167:
7163:
7161:
7158:
7157:
7141:
7138:
7137:
7121:
7118:
7117:
7100:
7096:
7094:
7091:
7090:
7064:
7061:
7060:
7035:
7032:
7031:
7015:
7012:
7011:
6994:
6990:
6982:
6979:
6978:
6968:
6946:
6943:
6942:
6917:
6914:
6913:
6891:
6888:
6887:
6879:{\textstyle 2k}
6868:
6865:
6864:
6848:
6845:
6844:
6818:
6814:
6812:
6809:
6808:
6786:
6783:
6782:
6766:
6763:
6762:
6746:
6743:
6742:
6726:
6723:
6722:
6721:with parameter
6699:
6696:
6695:
6679:
6676:
6675:
6641:
6638:
6637:
6629:{\textstyle np}
6618:
6615:
6614:
6579:
6576:
6575:
6568:
6553:{\textstyle na}
6542:
6539:
6538:
6522:
6519:
6518:
6517:for the sum of
6493:
6490:
6489:
6474:
6469:
6464:
6435:
6420:
6416:
6410:
6398:
6395:
6394:
6374:
6370:
6356:
6353:
6352:
6335:
6331:
6316:
6312:
6310:
6307:
6306:
6280:
6276:
6274:
6271:
6270:
6247:
6243:
6241:
6238:
6237:
6225:{\textstyle -1}
6214:
6211:
6210:
6153:
6140:
6135:
6126:
6115:
6111:
6106:
6105:
6101:
6100:
6088:
6083:
6065:
6061:
6059:
6058:
6046:
6045:
6039:
6034:
6021:
6017:
6006:
6002:
6000:
5999:
5997:
5996:
5992:
5986:
5985:
5976:
5972:
5957:
5953:
5945:
5942:
5941:
5925:
5922:
5921:
5901:
5896:
5883:
5879:
5865:
5862:
5861:
5841:
5837:
5820:
5817:
5816:
5799:
5795:
5780:
5776:
5774:
5771:
5770:
5767:conjugate prior
5746:
5745:
5736:
5732:
5728:
5723:
5721:
5715:
5714:
5709:
5701:
5697:
5692:
5685:
5684:
5672:
5668:
5653:
5652:
5650:
5647:
5646:
5629:
5625:
5623:
5620:
5619:
5603:
5600:
5599:
5568:
5563:
5550:
5545:
5540:
5533:
5529:
5523:
5519:
5510:
5506:
5502:
5500:
5490:
5486:
5482:
5466:
5461:
5448:
5443:
5438:
5431:
5427:
5421:
5417:
5413:
5410:
5392:
5388:
5379:
5375:
5366:
5362:
5360:
5357:
5356:
5328:
5323:
5313:
5308:
5302:
5279:
5274:
5264:
5259:
5253:
5252:
5248:
5238:
5226:
5221:
5213:
5206:
5202:
5196:
5192:
5183:
5179:
5175:
5173:
5161:
5157:
5148:
5144:
5131:
5130:
5126:
5124:
5121:
5120:
5100:
5095:
5082:
5078:
5063:
5059:
5057:
5054:
5053:
5033:
5028:
5015:
5011:
4996:
4992:
4990:
4987:
4986:
4959:
4956:
4955:
4939:
4936:
4935:
4907:
4904:
4903:
4887:
4884:
4883:
4860:
4857:
4856:
4840:
4837:
4836:
4835:) asserts that
4799:
4796:
4795:
4748:
4744:
4742:
4739:
4738:
4737:is of the form
4722:
4719:
4718:
4701:
4697:
4695:
4692:
4691:
4686:
4681:
4679:Maximum entropy
4676:
4671:
4666:
4661:
4637:
4633:
4628:
4625:
4624:
4606:
4602:
4590:
4586:
4580:
4576:
4564:
4560:
4554:
4550:
4538:
4534:
4528:
4524:
4512:
4508:
4506:
4503:
4502:
4480:
4477:
4476:
4458:
4454:
4439:
4435:
4429:
4425:
4413:
4409:
4403:
4399:
4387:
4383:
4381:
4378:
4377:
4354:
4350:
4345:
4342:
4341:
4323:
4319:
4307:
4303:
4297:
4293:
4281:
4277:
4271:
4267:
4255:
4251:
4249:
4246:
4245:
4223:
4220:
4219:
4201:
4197:
4182:
4178:
4172:
4168:
4156:
4152:
4150:
4147:
4146:
4123:
4119:
4114:
4111:
4110:
4092:
4088:
4076:
4072:
4066:
4062:
4050:
4046:
4044:
4041:
4040:
4018:
4015:
4014:
3996:
3992:
3977:
3973:
3971:
3968:
3967:
3944:
3940:
3938:
3935:
3934:
3916:
3912:
3903:
3899:
3897:
3894:
3893:
3871:
3868:
3867:
3850:
3847:
3846:
3838:Central moment
3827:
3821:
3796:
3773:
3770:
3769:
3752:
3748:
3746:
3743:
3742:
3726:
3723:
3722:
3706:
3703:
3702:
3691:
3662:
3658:
3656:
3653:
3652:
3608:
3604:
3598:
3594:
3561:
3557:
3555:
3552:
3551:
3547:
3507:
3503:
3480:
3478:
3475:
3474:
3420:
3418:
3415:
3414:
3393:Its density is
3361:
3358:
3357:
3329:
3326:
3325:
3309:
3306:
3305:
3265:
3261:
3254:
3250:
3241:
3237:
3224:
3222:
3202:
3200:
3197:
3196:
3160:
3156:
3145:
3143:
3120:
3118:
3115:
3114:
3095:
3092:
3091:
3064:
3061:
3060:
3035:
3032:
3031:
3006:
3003:
3002:
2954:
2951:
2950:
2946:
2941:
2929:
2926:
2923:
2921:
2912:
2909:
2906:
2904:
2893:
2890:
2887:
2885:
2876:
2873:
2870:
2868:
2857:
2854:
2851:
2849:
2840:
2837:
2834:
2832:
2821:
2818:
2815:
2813:
2804:
2801:
2798:
2796:
2785:
2782:
2779:
2777:
2768:
2765:
2762:
2760:
2749:
2746:
2743:
2741:
2732:
2729:
2726:
2724:
2713:
2710:
2707:
2705:
2696:
2693:
2690:
2688:
2665:
2661:
2659:
2656:
2655:
2638:
2635:
2634:
2613:
2609:
2607:
2604:
2603:
2586:
2583:
2582:
2577:
2571:
2556:
2550:
2547:
2545:
2535:
2532:
2529:
2527:
2521:
2518:
2515:
2513:
2498:
2496:
2490:
2487:
2485:
2475:
2472:
2469:
2467:
2461:
2458:
2455:
2453:
2438:
2436:
2430:
2428:
2418:
2415:
2412:
2410:
2404:
2401:
2398:
2396:
2380:
2370:
2367:
2365:
2360:
2350:
2347:
2344:
2342:
2336:
2333:
2330:
2328:
2312:
2302:
2299:
2297:
2292:
2282:
2279:
2276:
2274:
2268:
2265:
2262:
2260:
2244:
2234:
2231:
2228:
2226:
2221:
2211:
2208:
2205:
2203:
2197:
2194:
2191:
2189:
2147:
2139:
2137:
2134:
2133:
2110:
2107:
2106:
2041:
2038:
2037:
2020:
2017:
2016:
2002:
1992:
1967:
1963:
1955:
1952:
1951:
1934:
1930:
1928:
1925:
1924:
1899:
1896:
1895:
1872:
1868:
1853:
1849:
1835:
1832:
1831:
1828:
1823:
1818:
1813:
1808:
1803:
1798:
1793:
1788:
1779:
1627:Random variable
1578:Bernoulli trial
1433:
1419:
1394:
1392:
1388:
1382:
1378:
1377:
1373:
1372:
1368:
1366:
1362:
1347:
1346:
1344:
1333:
1330:
1329:
1297:
1292:
1282:
1277:
1271:
1248:
1243:
1232:
1228:
1222:
1218:
1209:
1205:
1201:
1199:
1190:
1178:
1174:
1168:
1164:
1162:
1158:
1157:
1156:
1152:
1142:
1140:
1137:
1136:
1110:
1109:
1100:
1096:
1085:
1080:
1074:
1073:
1068:
1062:
1058:
1053:
1043:
1042:
1030:
1026:
1011:
1010:
1008:
1005:
1004:
974:
968:
964:
958:
954:
931:
928:
927:
897:
891:
887:
881:
877:
857:
854:
853:
825:
821:
793:
791:
788:
787:
763:
760:
759:
755:Excess kurtosis
735:
732:
731:
702:
697:
692:
689:
688:
663:
659:
657:
654:
653:
629:
626:
625:
601:
598:
597:
573:
570:
569:
520:
516:
509:
498:
495:
494:
456:
452:
441:
439:
435:
422:
418:
408:
385:
383:
379:
374:
371:
370:
340:
336:
332:
325:
321:
308:
306:
302:
298:
289:
285:
273:
271:
268:
267:
243:
235:
232:
231:
221:
193:
188:
187:
178:
174:
172:
169:
168:
167:
143:
135:
132:
131:
103:
99:
84:
83:
81:
78:
77:
64:
57:
45:
35:
28:
23:
22:
15:
12:
11:
5:
19916:
19906:
19905:
19900:
19895:
19890:
19885:
19880:
19863:
19862:
19860:
19859:
19849:
19838:
19835:
19834:
19832:
19831:
19826:
19821:
19816:
19811:
19806:
19804:Locationâscale
19801:
19796:
19791:
19786:
19781:
19775:
19773:
19769:
19768:
19766:
19765:
19760:
19753:
19748:
19740:
19738:
19727:
19726:
19724:
19723:
19718:
19713:
19706:
19701:
19694:
19689:
19682:
19677:
19672:
19667:
19665:Wrapped Cauchy
19662:
19660:Wrapped normal
19657:
19652:
19647:
19636:
19634:
19628:
19627:
19625:
19624:
19623:
19622:
19617:
19615:Normal-inverse
19612:
19607:
19597:
19596:
19595:
19585:
19577:
19572:
19567:
19558:
19557:
19556:
19546:
19538:
19533:
19528:
19523:
19522:
19521:
19511:
19504:
19503:
19502:
19497:
19487:
19482:
19474:
19472:
19464:
19463:
19460:
19459:
19457:
19456:
19450:
19448:
19439:
19433:
19432:
19429:
19428:
19426:
19425:
19420:
19415:
19407:
19399:
19391:
19382:
19373:
19364:
19355:
19346:
19341:
19336:
19331:
19325:
19323:
19317:
19316:
19314:
19313:
19308:
19306:Variance-gamma
19303:
19298:
19290:
19285:
19280:
19275:
19270:
19265:
19257:
19252:
19251:
19250:
19240:
19235:
19230:
19224:
19219:
19214:
19209:
19204:
19199:
19194:
19186:
19181:
19173:
19168:
19162:
19160:
19152:
19151:
19149:
19148:
19146:Wilks's lambda
19143:
19142:
19141:
19131:
19126:
19121:
19116:
19111:
19106:
19101:
19096:
19091:
19086:
19084:Mittag-Leffler
19081:
19076:
19071:
19066:
19061:
19056:
19051:
19046:
19041:
19036:
19031:
19026:
19025:
19024:
19014:
19005:
19000:
18995:
18994:
18993:
18983:
18981:gamma/Gompertz
18978:
18977:
18976:
18971:
18961:
18956:
18951:
18950:
18949:
18937:
18936:
18935:
18930:
18925:
18915:
18914:
18913:
18903:
18898:
18893:
18892:
18891:
18890:
18889:
18879:
18869:
18864:
18859:
18854:
18849:
18844:
18838:
18836:
18833:semi-infinite
18828:
18827:
18825:
18824:
18819:
18814:
18809:
18804:
18799:
18794:
18789:
18784:
18779:
18774:
18769:
18764:
18759:
18754:
18749:
18744:
18739:
18733:
18731:
18722:
18716:
18715:
18712:
18711:
18709:
18708:
18703:
18698:
18693:
18688:
18683:
18678:
18673:
18668:
18663:
18658:
18653:
18648:
18643:
18638:
18633:
18628:
18623:
18617:
18615:
18612:with infinite
18609:
18608:
18606:
18605:
18600:
18595:
18590:
18585:
18580:
18575:
18574:
18573:
18566:Hypergeometric
18563:
18558:
18553:
18548:
18543:
18537:
18535:
18526:
18520:
18519:
18507:
18506:
18499:
18492:
18484:
18478:
18477:
18472:
18452:
18451:External links
18449:
18446:
18445:
18439:
18427:Abramowitz, M.
18410:
18387:
18372:
18352:(1): 119â127.
18338:Wallace, C. S.
18334:
18328:
18308:
18302:
18289:
18283:
18266:
18248:(2): 137â138.
18237:
18217:(2): 239â265.
18199:
18181:(3): 323â333.
18170:
18152:(4): 357â372.
18141:
18123:(3): 507â513.
18112:
18094:(2): 108â114.
18083:
18071:(3): 303â322.
18056:
18023:
18005:(1): 169â212.
17987:
17946:
17940:
17927:
17900:
17890:(124): 19â32.
17876:
17849:
17819:
17813:
17796:
17776:(2): 389â394.
17761:
17748:
17713:(4): 449â453.
17695:
17689:
17676:
17659:
17627:
17621:
17608:
17602:
17589:
17569:(3): 257â260.
17554:
17515:
17509:
17496:
17490:
17477:
17463:
17450:
17444:
17421:
17403:
17397:
17384:
17355:
17349:
17331:
17309:
17303:
17286:
17272:
17237:
17231:
17213:
17184:
17175:
17148:
17142:
17129:
17123:
17110:
17104:
17091:
17085:
17072:
17046:
17036:
17035:
17033:
17030:
17027:
17026:
16979:
16965:
16953:
16938:
16921:
16909:
16907:, p. 244)
16897:
16895:, p. 243)
16885:
16883:, p. 144)
16873:
16871:, p. 189)
16861:
16859:, Problem III)
16849:
16847:, section 179)
16837:
16835:, section 177)
16825:
16813:
16800:
16788:
16776:
16773:Wallace (1996)
16765:
16753:
16742:
16731:
16720:
16709:
16697:
16662:
16653:
16646:
16621:
16614:
16594:
16583:
16581:, p. 133)
16571:
16569:, p. 130)
16559:
16557:, p. 127)
16544:
16533:(8): 879â885.
16517:
16494:
16480:
16468:
16456:
16449:
16431:
16394:
16343:
16340:. wolfram.com.
16324:
16312:
16250:
16225:
16200:
16189:
16171:
16145:
16130:
16115:
16100:
16098:(Section 5.40)
16079:
16058:
16032:
16025:
15999:
15997:, p. 254)
15987:
15975:
15963:
15942:
15927:
15915:
15903:
15886:
15823:Lukacs, Eugene
15814:
15805:
15769:(2): 219â230.
15746:
15739:
15713:
15706:
15678:
15647:
15616:
15591:
15566:
15547:
15545:, p. 121)
15535:
15533:, p. 110)
15523:
15511:
15508:Stigler (1982)
15500:
15472:
15465:
15447:
15434:
15432:, p. 102)
15422:
15408:
15348:
15347:
15345:
15342:
15340:
15337:
15334:
15333:
15300:
15283:
15270:
15236:
15222:
15221:
15219:
15216:
15215:
15214:
15208:
15202:
15192:
15187:
15182:
15177:
15172:
15167:
15162:
15157:
15140:
15136:
15133:
15126:
15123:
15120:
15117:
15114:
15111:
15110:
15106:
15100:
15097:
15092:
15089:
15085:
15081:
15080:
15076:
15070:
15066:
15060:
15053:
15050:
15047:
15044:
15041:
15038:
15035:
15010:
15003:
14999:
14994:
14989:
14986:
14980:
14975:
14970:
14967:
14964:
14961:
14956:
14952:
14948:
14945:
14942:
14939:
14936:
14931:
14928:
14925:
14921:
14914:
14911:
14906:
14902:
14896:
14893:
14890:
14887:
14884:
14864:
14861:
14858:
14855:
14852:
14838:
14828:
14823:
14814:
14808:
14802:
14795:
14794:
14778:
14775:
14770:Pearson (1920)
14766:
14763:
14760:
14728:
14725:
14723:
14720:
14719:
14718:
14706:
14703:
14700:
14672:
14667:
14663:
14657:
14653:
14648:
14642:
14639:
14636:
14633:
14630:
14625:
14621:
14615:
14612:
14609:
14606:
14603:
14598:
14594:
14587:
14582:
14576:
14573:
14570:
14567:
14564:
14559:
14555:
14549:
14546:
14543:
14540:
14537:
14532:
14528:
14521:
14517:
14515:
14511:
14505:
14502:
14499:
14496:
14493:
14488:
14484:
14478:
14475:
14472:
14469:
14466:
14461:
14457:
14450:
14445:
14439:
14436:
14433:
14430:
14427:
14422:
14418:
14412:
14409:
14406:
14403:
14400:
14395:
14391:
14384:
14380:
14378:
14374:
14368:
14365:
14362:
14359:
14356:
14351:
14347:
14341:
14338:
14335:
14332:
14329:
14324:
14320:
14313:
14308:
14302:
14299:
14296:
14292:
14287:
14283:
14280:
14278:
14275:
14272:
14269:
14265:
14262:
14259:
14256:
14255:
14236:
14233:
14230:
14209:
14203:
14200:
14196:
14192:
14189:
14186:
14182:
14160:
14157:
14153:
14132:
14129:
14126:
14112:
14101:
14073:
14069:
14066:
14060:
14057:
14054:
14051:
14048:
14045:
14042:
14036:
14032:
14026:
14020:
14017:
14014:
14011:
14008:
14002:
13998:
13992:
13986:
13983:
13980:
13974:
13970:
13964:
13959:
13954:
13950:
13944:
13941:
13937:
13933:
13930:
13927:
13924:
13921:
13916:
13913:
13908:
13905:
13902:
13899:
13896:
13882:
13872:
13854:
13853:= 1.330274429.
13850:
13843:
13836:
13829:
13822:
13815:
13792:
13786:
13781:
13777:
13773:
13770:
13766:
13761:
13758:
13754:
13751:
13748:
13745:
13742:
13739:
13735:
13729:
13725:
13719:
13715:
13711:
13706:
13702:
13696:
13692:
13688:
13683:
13679:
13673:
13669:
13665:
13660:
13656:
13650:
13646:
13642:
13639:
13634:
13630:
13625:
13621:
13618:
13615:
13612:
13609:
13606:
13603:
13600:
13597:
13594:
13591:
13548:
13545:
13544:
13543:
13535:
13528:
13520:
13519:
13518:
13514:
13513:
13498:
13479:
13461:
13436:
13422:
13399:
13395:
13392:
13389:
13386:
13383:
13376:
13373:
13370:
13366:
13360:
13356:
13353:
13350:
13347:
13344:
13337:
13334:
13331:
13290:
13273:will have the
13236:
13233:
13230:
13227:
13224:
13221:
13218:
13215:
13209:
13206:
13203:
13200:
13197:
13192:
13189:
13185:
13182:
13179:
13176:
13173:
13170:
13167:
13164:
13158:
13155:
13152:
13149:
13146:
13141:
13138:
13104:
13086:
13050:Francis Galton
13037:
13034:
13032:
13029:
13027:
13024:
13023:
13022:
12991:
12948:
12947:
12932:
12925:
12890:
12889:
12888:
12885:
12870:
12847:Pearson (1901)
12843:
12835:from Fisher's
12824:
12821:
12820:
12819:
12809:
12808:
12807:
12801:
12782:
12779:
12778:
12777:
12753:
12750:
12747:
12744:
12724:
12721:
12718:
12715:
12712:
12709:
12701:
12697:
12693:
12687:
12683:
12675:
12672:
12667:
12664:
12661:
12658:
12655:
12652:
12649:
12643:
12640:
12636:
12603:
12582:
12579:
12578:
12577:
12574:
12567:
12560:
12555:
12552:
12547:
12546:
12545:
12526:
12514:
12510:
12506:
12499:
12490:
12487:
12485:
12482:
12480:
12477:
12475:
12472:
12470:
12467:
12466:
12465:
12438:
12435:
12432:
12427:
12424:
12392:of quantities
12377:
12372:
12369:
12365:
12359:
12356:
12352:
12348:
12343:
12340:
12336:
12332:
12329:
12321:
12318:
12313:
12308:
12305:
12299:
12294:
12288:
12285:
12282:
12277:
12274:
12261:
12234:
12231:
12228:
12223:
12220:
12217:
12214:
12211:
12195:
12192:
12190:
12187:
12186:
12185:
12170:
12156:
12149:
12142:
12137:
12134:
12133:
12132:
12122:
12116:
12115:
12105:
12100:
12094:
12093:
12077:
12074:
12067:
12063:
12057:
12054:
12048:
12043:
12040:
12037:
12033:
12029:
12026:
12021:
12018:
12015:
12011:
11996:
11986:
11977:
11956: = (
11952:
11945:
11935:
11927:
11906:Main article:
11903:
11900:
11893:Studentization
11888:
11885:
11873:
11870:
11862:
11859:
11847:
11844:
11842:
11839:
11838:
11837:
11830:
11823:
11822:
11803:
11784:
11774:
11773:
11772:
11766:
11760:
11733:scalar product
11687:
11681:
11645:
11639:
11601:
11597:
11587:
11555:
11552:
11550:
11547:
11545:
11542:
11540:
11537:
11536:
11535:
11519:
11516:
11511:
11507:
11503:
11500:
11495:
11490:
11486:
11482:
11479:
11476:
11465:quadratic form
11459:
11456:
11455:
11454:
11443:
11438:
11435:
11432:
11428:
11424:
11418:
11414:
11409:
11403:
11398:
11394:
11390:
11387:
11384:
11379:
11374:
11370:
11366:
11361:
11356:
11352:
11347:
11341:
11337:
11332:
11326:
11321:
11317:
11313:
11310:
11307:
11302:
11297:
11293:
11289:
11284:
11279:
11275:
11270:
11263:
11260:
11236:F-distribution
11220:
11216:
11212:
11209:
11206:
11201:
11197:
11193:
11188:
11184:
11161:
11157:
11153:
11150:
11147:
11142:
11138:
11134:
11129:
11125:
11113:
11102:
11097:
11094:
11091:
11087:
11083:
11076:
11070:
11066:
11060:
11057:
11052:
11047:
11043:
11039:
11036:
11033:
11030:
11025:
11021:
11015:
11012:
11007:
11002:
10998:
10994:
10990:
10983:
10980:
10977:
10974:
10971:
10968:
10964:
10957:
10954:
10951:
10946:
10942:
10938:
10935:
10932:
10927:
10923:
10919:
10914:
10911:
10903:
10895:
10889:
10885:
10880:
10877:
10872:
10869:
10861:
10858:
10838:
10835:
10832:
10813:Basu's theorem
10790:
10786:
10776:and variances
10765:
10743:
10739:
10735:
10732:
10729:
10724:
10720:
10716:
10711:
10707:
10695:
10684:
10679:
10674:
10670:
10666:
10661:
10656:
10652:
10648:
10645:
10642:
10637:
10632:
10628:
10611:{\textstyle n}
10607:
10581:
10577:
10573:
10570:
10567:
10562:
10558:
10554:
10549:
10545:
10533:
10524:
10521:
10520:
10519:
10499:
10494:
10490:
10486:
10481:
10476:
10472:
10458:
10446:
10443:
10440:
10437:
10434:
10431:
10428:
10425:
10420:
10416:
10411:
10405:
10401:
10385:
10371:
10367:
10363:
10360:
10356:
10350:
10346:
10342:
10339:
10336:
10333:
10330:
10327:
10324:
10319:
10315:
10302:, and has the
10291:
10288:
10285:
10259:
10255:
10234:
10230:
10226:
10222:
10218:
10213:
10209:
10203:
10200:
10196:
10192:
10189:
10186:
10183:
10178:
10174:
10147:
10143:
10137:
10133:
10129:
10126:
10117:Their product
10115:
10103:
10100:
10097:
10094:
10091:
10086:
10081:
10076:
10072:
10068:
10063:
10059:
10045:
10042:
10041:
10040:
10019:
10014:
10010:
10006:
10003:
10000:
9997:
9994:
9991:
9986:
9981:
9977:
9973:
9966:
9961:
9957:
9951:
9946:
9942:
9935:
9930:
9925:
9921:
9895:
9890:
9886:
9882:
9879:
9876:
9873:
9870:
9867:
9862:
9857:
9853:
9849:
9842:
9837:
9833:
9827:
9823:
9819:
9816:
9813:
9810:
9807:
9804:
9799:
9794:
9790:
9784:
9780:
9776:
9770:
9765:
9761:
9740:
9735:
9730:
9726:
9722:
9717:
9713:
9709:
9704:
9680:
9677:
9674:
9669:
9665:
9659:
9654:
9650:
9643:
9634:
9631:
9628:
9623:
9620:
9617:
9612:
9608:
9604:
9601:
9598:
9593:
9588:
9584:
9576:
9571:
9565:
9560:
9548:geometric mean
9535:
9532:
9529:
9526:
9523:
9520:
9517:
9497:
9492:
9487:
9483:
9479:
9474:
9470:
9466:
9461:
9456:
9451:
9447:
9435:
9423:
9420:
9417:
9391:
9387:
9366:
9346:
9343:
9335:
9331:
9327:
9322:
9318:
9312:
9309:
9306:
9303:
9300:
9297:
9294:
9289:
9285:
9281:
9278:
9273:
9269:
9265:
9259:
9254:
9250:
9233:{\textstyle b}
9229:
9213:{\textstyle a}
9209:
9187:
9183:
9162:
9140:
9136:
9113:
9109:
9097:
9079:
9075:
9071:
9051:
9048:
9045:
9025:
9022:
9019:
8997:
8993:
8976:{\textstyle Y}
8972:
8956:{\textstyle X}
8952:
8941:
8927:
8922:
8918:
8914:
8909:
8904:
8900:
8877:
8873:
8869:
8864:
8860:
8837:
8833:
8829:
8824:
8820:
8797:
8792:
8788:
8765:
8760:
8756:
8746:and variances
8733:
8729:
8706:
8702:
8675:
8671:
8648:
8644:
8630:
8627:
8626:
8625:
8611:
8608:
8604:
8577:
8574:
8570:
8566:
8563:
8560:
8557:
8547:
8542:is called the
8531:
8528:
8525:
8522:
8519:
8503:{\textstyle X}
8499:
8488:
8472:
8468:
8462:
8459:
8454:
8450:
8446:
8443:
8440:
8435:
8430:
8425:
8421:
8418:
8415:
8409:
8402:
8399:
8394:
8391:
8388:
8385:
8382:
8379:
8376:
8373:
8353:{\textstyle x}
8349:
8338:
8322:
8319:
8316:
8296:
8291:
8287:
8282:
8276:
8272:
8268:
8263:
8258:
8254:
8250:
8245:
8241:
8236:
8230:
8226:
8201:
8197:
8193:
8184:The square of
8182:
8168:
8164:
8160:
8157:
8153:
8148:
8144:
8141:
8138:
8134:
8109:
8105:
8100:
8096:
8093:
8090:
8086:
8074:
8058:
8055:
8052:
8031:
8026:
8022:
8018:
8015:
8012:
8007:
8003:
7999:
7995:
7992:
7989:
7967:{\textstyle X}
7963:
7952:
7940:
7937:
7932:
7928:
7923:
7920:
7917:
7912:
7907:
7904:
7901:
7898:
7895:
7892:
7889:
7869:{\textstyle X}
7865:
7850:
7838:
7835:
7830:
7826:
7822:
7819:
7816:
7813:
7810:
7807:
7804:
7801:
7796:
7792:
7771:{\textstyle X}
7767:
7756:
7738:
7734:
7728:
7724:
7703:
7700:
7697:
7694:
7678:{\textstyle b}
7674:
7658:{\textstyle a}
7654:
7634:
7631:
7628:
7625:
7613:
7610:
7596:
7591:
7587:
7583:
7578:
7573:
7570:
7567:
7563:
7538:
7535:
7530:
7527:
7523:
7502:
7499:
7494:
7489:
7485:
7464:
7461:
7456:
7451:
7447:
7426:
7423:
7418:
7414:
7393:
7390:
7387:
7382:
7378:
7361:{\textstyle y}
7357:
7341:{\textstyle x}
7337:
7313:
7310:
7305:
7302:
7298:
7277:
7274:
7269:
7265:
7244:
7241:
7236:
7232:
7211:
7208:
7203:
7199:
7178:
7175:
7170:
7166:
7149:{\textstyle y}
7145:
7129:{\textstyle x}
7125:
7103:
7099:
7074:
7071:
7068:
7048:
7045:
7042:
7039:
7023:{\textstyle x}
7019:
6997:
6993:
6989:
6986:
6967:
6964:
6963:
6962:
6950:
6930:
6927:
6924:
6921:
6907:
6899:{\textstyle k}
6895:
6875:
6872:
6856:{\textstyle k}
6852:
6832:
6829:
6826:
6821:
6817:
6802:
6790:
6770:
6750:
6730:
6715:
6707:{\textstyle p}
6703:
6687:{\textstyle n}
6683:
6663:
6660:
6657:
6654:
6651:
6648:
6645:
6625:
6622:
6598:
6595:
6592:
6589:
6586:
6583:
6564:Main article:
6549:
6546:
6530:{\textstyle n}
6526:
6506:
6503:
6500:
6497:
6473:
6470:
6468:
6465:
6463:
6462:
6448:
6445:
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6431:
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6423:
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6413:
6409:
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6382:
6377:
6373:
6369:
6366:
6363:
6360:
6338:
6334:
6330:
6327:
6324:
6319:
6315:
6303:
6289:
6286:
6283:
6279:
6256:
6253:
6250:
6246:
6221:
6218:
6196:
6177:
6165:
6159:
6156:
6151:
6143:
6138:
6134:
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6118:
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6110:
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6091:
6086:
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6078:
6073:
6068:
6064:
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6042:
6037:
6033:
6029:
6024:
6020:
6014:
6009:
6005:
5995:
5989:
5984:
5979:
5975:
5971:
5968:
5965:
5960:
5956:
5952:
5949:
5929:
5909:
5904:
5899:
5895:
5891:
5886:
5882:
5878:
5875:
5872:
5869:
5849:
5844:
5840:
5836:
5833:
5830:
5827:
5824:
5802:
5798:
5794:
5791:
5788:
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5779:
5763:
5750:
5739:
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5727:
5722:
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5710:
5704:
5700:
5696:
5691:
5690:
5688:
5683:
5680:
5675:
5671:
5667:
5664:
5661:
5656:
5632:
5628:
5607:
5592:
5580:
5571:
5566:
5562:
5558:
5553:
5548:
5544:
5536:
5532:
5526:
5522:
5518:
5513:
5509:
5505:
5497:
5494:
5489:
5485:
5481:
5478:
5469:
5464:
5460:
5456:
5451:
5446:
5442:
5434:
5430:
5424:
5420:
5416:
5409:
5406:
5403:
5400:
5395:
5391:
5387:
5382:
5378:
5374:
5369:
5365:
5339:
5331:
5326:
5322:
5316:
5311:
5307:
5301:
5298:
5295:
5292:
5289:
5282:
5277:
5273:
5267:
5262:
5258:
5251:
5245:
5242:
5237:
5229:
5224:
5220:
5216:
5209:
5205:
5199:
5195:
5191:
5186:
5182:
5178:
5172:
5169:
5164:
5160:
5156:
5151:
5147:
5143:
5137:
5134:
5129:
5108:
5103:
5098:
5094:
5090:
5085:
5081:
5077:
5074:
5071:
5066:
5062:
5041:
5036:
5031:
5027:
5023:
5018:
5014:
5010:
5007:
5004:
4999:
4995:
4979:
4967:{\textstyle Y}
4963:
4947:{\textstyle X}
4943:
4924:jointly normal
4915:{\textstyle Y}
4911:
4895:{\textstyle X}
4891:
4880:
4868:{\textstyle X}
4864:
4848:{\textstyle Q}
4844:
4812:
4809:
4806:
4803:
4783:
4780:
4777:
4774:
4771:
4768:
4765:
4762:
4759:
4756:
4751:
4747:
4730:{\textstyle X}
4726:
4704:
4700:
4687:
4685:
4682:
4680:
4677:
4675:
4672:
4670:
4667:
4665:
4662:
4660:
4657:
4654:
4653:
4640:
4636:
4632:
4622:
4609:
4605:
4601:
4598:
4593:
4589:
4583:
4579:
4575:
4572:
4567:
4563:
4557:
4553:
4549:
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4541:
4537:
4531:
4527:
4523:
4520:
4515:
4511:
4500:
4496:
4495:
4488:{\textstyle 0}
4484:
4474:
4461:
4457:
4453:
4450:
4447:
4442:
4438:
4432:
4428:
4424:
4421:
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4412:
4406:
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4398:
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4390:
4386:
4375:
4371:
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4353:
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4326:
4322:
4318:
4315:
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4306:
4300:
4296:
4292:
4289:
4284:
4280:
4274:
4270:
4266:
4263:
4258:
4254:
4243:
4239:
4238:
4231:{\textstyle 0}
4227:
4217:
4204:
4200:
4196:
4193:
4190:
4185:
4181:
4175:
4171:
4167:
4164:
4159:
4155:
4144:
4140:
4139:
4126:
4122:
4118:
4108:
4095:
4091:
4087:
4084:
4079:
4075:
4069:
4065:
4061:
4058:
4053:
4049:
4038:
4034:
4033:
4026:{\textstyle 0}
4022:
4012:
3999:
3995:
3991:
3988:
3985:
3980:
3976:
3965:
3961:
3960:
3947:
3943:
3932:
3919:
3915:
3911:
3906:
3902:
3891:
3887:
3886:
3879:{\textstyle 0}
3875:
3865:
3854:
3844:
3840:
3839:
3836:
3833:
3820:
3817:
3816:
3815:
3803:
3799:
3795:
3792:
3789:
3786:
3783:
3780:
3777:
3755:
3751:
3730:
3714:{\textstyle X}
3710:
3699:
3679:
3676:
3673:
3670:
3665:
3661:
3640:
3637:
3634:
3631:
3628:
3625:
3622:
3619:
3616:
3611:
3607:
3601:
3597:
3593:
3590:
3587:
3584:
3581:
3578:
3575:
3570:
3567:
3564:
3560:
3544:
3533:
3530:
3527:
3524:
3521:
3518:
3515:
3510:
3506:
3502:
3499:
3496:
3493:
3490:
3486:
3483:
3471:
3460:
3457:
3454:
3451:
3448:
3445:
3442:
3439:
3436:
3433:
3430:
3426:
3423:
3410:
3409:
3402:differentiable
3398:
3391:
3380:
3377:
3374:
3371:
3368:
3365:
3345:
3342:
3339:
3336:
3333:
3317:{\textstyle f}
3313:
3298:
3287:
3284:
3281:
3278:
3275:
3268:
3264:
3257:
3253:
3249:
3244:
3240:
3236:
3233:
3230:
3227:
3221:
3218:
3215:
3212:
3208:
3205:
3193:
3182:
3179:
3176:
3173:
3170:
3163:
3159:
3154:
3151:
3148:
3142:
3139:
3136:
3133:
3130:
3126:
3123:
3111:
3103:{\textstyle x}
3099:
3088:
3077:
3074:
3071:
3068:
3048:
3045:
3042:
3039:
3019:
3016:
3013:
3010:
2991:
2967:
2964:
2961:
2958:
2945:
2942:
2940:
2937:
2934:
2933:
2919:
2916:
2902:
2898:
2897:
2883:
2880:
2866:
2862:
2861:
2847:
2844:
2830:
2826:
2825:
2811:
2808:
2794:
2790:
2789:
2775:
2772:
2758:
2754:
2753:
2739:
2736:
2722:
2718:
2717:
2703:
2700:
2686:
2682:
2681:
2668:
2664:
2653:
2646:{\textstyle p}
2642:
2632:
2629:
2616:
2612:
2601:
2594:{\textstyle p}
2590:
2570:
2567:
2564:
2563:
2560:
2559:
2554:
2539:
2525:
2511:
2507:
2506:
2503:
2502:
2494:
2479:
2465:
2451:
2447:
2446:
2443:
2442:
2434:
2422:
2408:
2394:
2390:
2389:
2378:
2375:
2374:
2363:
2354:
2340:
2326:
2322:
2321:
2310:
2307:
2306:
2295:
2286:
2272:
2258:
2254:
2253:
2242:
2239:
2238:
2224:
2215:
2201:
2187:
2183:
2182:
2177:
2166:
2163:
2160:
2157:
2154:
2149: in
2146:
2131:
2120:
2117:
2114:
2104:
2093:
2090:
2087:
2084:
2081:
2078:
2075:
2072:
2069:
2066:
2063:
2060:
2057:
2054:
2051:
2048:
2045:
2035:
2028:{\textstyle n}
2024:
1991:
1988:
1987:
1986:
1975:
1970:
1966:
1962:
1959:
1937:
1933:
1912:
1909:
1906:
1903:
1883:
1880:
1875:
1871:
1867:
1864:
1861:
1856:
1852:
1848:
1845:
1842:
1839:
1827:
1824:
1822:
1819:
1817:
1816:Error function
1814:
1812:
1809:
1807:
1804:
1802:
1799:
1797:
1794:
1792:
1789:
1787:
1784:
1781:
1780:
1778:
1777:
1770:
1763:
1755:
1752:
1751:
1750:
1749:
1744:
1736:
1735:
1734:
1733:
1728:
1726:Bayes' theorem
1723:
1718:
1713:
1708:
1700:
1699:
1698:
1697:
1692:
1687:
1682:
1674:
1673:
1672:
1671:
1670:
1669:
1664:
1659:
1657:Observed value
1654:
1649:
1644:
1642:Expected value
1639:
1634:
1624:
1619:
1618:
1617:
1612:
1607:
1602:
1597:
1592:
1582:
1581:
1580:
1570:
1569:
1568:
1563:
1558:
1553:
1548:
1538:
1533:
1525:
1524:
1523:
1522:
1517:
1512:
1511:
1510:
1500:
1499:
1498:
1485:
1484:
1476:
1475:
1469:
1468:
1457:
1456:
1442:
1439:
1436:
1428:
1422:
1417:
1412:
1407:
1403:
1400:
1397:
1391:
1385:
1381:
1376:
1371:
1365:
1358:
1355:
1351:
1343:
1340:
1337:
1327:
1321:
1320:
1308:
1300:
1295:
1291:
1285:
1280:
1276:
1270:
1267:
1264:
1261:
1258:
1251:
1246:
1242:
1235:
1231:
1225:
1221:
1217:
1212:
1208:
1204:
1198:
1193:
1188:
1181:
1177:
1171:
1167:
1161:
1155:
1149:
1146:
1134:
1128:
1127:
1114:
1108:
1103:
1099:
1095:
1092:
1088:
1084:
1081:
1079:
1076:
1075:
1072:
1069:
1065:
1061:
1056:
1052:
1049:
1048:
1046:
1041:
1038:
1033:
1029:
1025:
1022:
1019:
1014:
1002:
996:
995:
984:
981:
977:
971:
967:
961:
957:
953:
950:
947:
944:
941:
938:
935:
925:
919:
918:
907:
904:
900:
894:
890:
884:
880:
876:
873:
870:
867:
864:
861:
851:
845:
844:
833:
828:
824:
820:
817:
814:
811:
808:
805:
800:
797:
785:
779:
778:
767:
757:
751:
750:
739:
729:
723:
722:
709:
705:
701:
696:
686:
680:
679:
666:
662:
651:
645:
644:
633:
623:
617:
616:
605:
595:
589:
588:
577:
567:
561:
560:
549:
546:
543:
540:
537:
534:
531:
526:
523:
519:
513:
508:
505:
502:
492:
486:
485:
473:
468:
460:
455:
450:
447:
444:
438:
434:
431:
428:
425:
421:
415:
412:
407:
403:
398:
394:
391:
388:
382:
378:
368:
362:
361:
343:
339:
335:
328:
324:
320:
317:
314:
311:
305:
301:
292:
288:
284:
281:
277:
265:
259:
258:
246:
242:
239:
229:
223:
222:
199:
196:
191:
186:
181:
177:
146:
142:
139:
129:
123:
122:
111:
106:
102:
98:
95:
92:
87:
75:
71:
70:
62:
59:
58:
51:
43:
26:
9:
6:
4:
3:
2:
19915:
19904:
19901:
19899:
19896:
19894:
19891:
19889:
19886:
19884:
19881:
19879:
19876:
19875:
19873:
19858:
19850:
19848:
19840:
19839:
19836:
19830:
19827:
19825:
19822:
19820:
19817:
19815:
19812:
19810:
19807:
19805:
19802:
19800:
19797:
19795:
19792:
19790:
19787:
19785:
19782:
19780:
19777:
19776:
19774:
19770:
19764:
19761:
19758:
19754:
19752:
19749:
19746:
19742:
19741:
19739:
19737:
19732:
19728:
19722:
19719:
19717:
19714:
19711:
19707:
19705:
19702:
19699:
19695:
19693:
19690:
19687:
19683:
19681:
19678:
19676:
19673:
19671:
19668:
19666:
19663:
19661:
19658:
19656:
19653:
19651:
19648:
19645:
19644:
19638:
19637:
19635:
19633:
19629:
19621:
19618:
19616:
19613:
19611:
19608:
19606:
19603:
19602:
19601:
19598:
19594:
19591:
19590:
19589:
19586:
19584:
19583:
19578:
19576:
19575:Matrix normal
19573:
19571:
19568:
19565:
19564:
19559:
19555:
19552:
19551:
19550:
19547:
19545:
19544:
19541:Multivariate
19539:
19537:
19534:
19532:
19529:
19527:
19524:
19520:
19517:
19516:
19515:
19512:
19509:
19505:
19501:
19498:
19496:
19493:
19492:
19491:
19488:
19486:
19483:
19480:
19476:
19475:
19473:
19471:
19468:Multivariate
19465:
19455:
19452:
19451:
19449:
19443:
19440:
19434:
19424:
19421:
19419:
19416:
19414:
19412:
19408:
19406:
19404:
19400:
19398:
19396:
19392:
19390:
19388:
19383:
19381:
19379:
19374:
19372:
19370:
19365:
19363:
19361:
19356:
19354:
19352:
19347:
19345:
19342:
19340:
19337:
19335:
19332:
19330:
19327:
19326:
19324:
19320:with support
19318:
19312:
19309:
19307:
19304:
19302:
19299:
19297:
19296:
19291:
19289:
19286:
19284:
19281:
19279:
19276:
19274:
19271:
19269:
19266:
19264:
19263:
19258:
19256:
19253:
19249:
19246:
19245:
19244:
19241:
19239:
19236:
19234:
19233:
19225:
19223:
19220:
19218:
19215:
19213:
19210:
19208:
19205:
19203:
19200:
19198:
19195:
19193:
19192:
19187:
19185:
19182:
19180:
19179:
19174:
19172:
19169:
19167:
19164:
19163:
19161:
19157:on the whole
19153:
19147:
19144:
19140:
19137:
19136:
19135:
19132:
19130:
19129:type-2 Gumbel
19127:
19125:
19122:
19120:
19117:
19115:
19112:
19110:
19107:
19105:
19102:
19100:
19097:
19095:
19092:
19090:
19087:
19085:
19082:
19080:
19077:
19075:
19072:
19070:
19067:
19065:
19062:
19060:
19057:
19055:
19052:
19050:
19047:
19045:
19042:
19040:
19037:
19035:
19032:
19030:
19027:
19023:
19020:
19019:
19018:
19015:
19013:
19011:
19006:
19004:
19001:
18999:
18998:Half-logistic
18996:
18992:
18989:
18988:
18987:
18984:
18982:
18979:
18975:
18972:
18970:
18967:
18966:
18965:
18962:
18960:
18957:
18955:
18954:Folded normal
18952:
18948:
18945:
18944:
18943:
18942:
18938:
18934:
18931:
18929:
18926:
18924:
18921:
18920:
18919:
18916:
18912:
18909:
18908:
18907:
18904:
18902:
18899:
18897:
18894:
18888:
18885:
18884:
18883:
18880:
18878:
18875:
18874:
18873:
18870:
18868:
18865:
18863:
18860:
18858:
18855:
18853:
18850:
18848:
18845:
18843:
18840:
18839:
18837:
18829:
18823:
18820:
18818:
18815:
18813:
18810:
18808:
18805:
18803:
18800:
18798:
18797:Raised cosine
18795:
18793:
18790:
18788:
18785:
18783:
18780:
18778:
18775:
18773:
18770:
18768:
18765:
18763:
18760:
18758:
18755:
18753:
18750:
18748:
18745:
18743:
18740:
18738:
18735:
18734:
18732:
18726:
18723:
18717:
18707:
18704:
18702:
18699:
18697:
18694:
18692:
18689:
18687:
18684:
18682:
18679:
18677:
18674:
18672:
18671:Mixed Poisson
18669:
18667:
18664:
18662:
18659:
18657:
18654:
18652:
18649:
18647:
18644:
18642:
18639:
18637:
18634:
18632:
18629:
18627:
18624:
18622:
18619:
18618:
18616:
18610:
18604:
18601:
18599:
18596:
18594:
18591:
18589:
18586:
18584:
18581:
18579:
18576:
18572:
18569:
18568:
18567:
18564:
18562:
18559:
18557:
18554:
18552:
18551:Beta-binomial
18549:
18547:
18544:
18542:
18539:
18538:
18536:
18530:
18527:
18521:
18516:
18512:
18505:
18500:
18498:
18493:
18491:
18486:
18485:
18482:
18476:
18473:
18469:
18465:
18464:
18459:
18455:
18454:
18442:
18436:
18432:
18431:Stegun, I. A.
18428:
18424:
18423:
18418:
18417:
18411:
18404:
18400:
18393:
18388:
18384:
18380:
18377:
18373:
18369:
18365:
18360:
18355:
18351:
18347:
18343:
18339:
18335:
18331:
18325:
18321:
18314:
18309:
18305:
18299:
18295:
18290:
18286:
18280:
18275:
18274:
18267:
18263:
18259:
18255:
18251:
18247:
18243:
18238:
18234:
18230:
18225:
18220:
18216:
18212:
18208:
18204:
18200:
18196:
18192:
18188:
18184:
18180:
18176:
18171:
18167:
18163:
18159:
18155:
18151:
18147:
18142:
18138:
18134:
18130:
18126:
18122:
18118:
18113:
18109:
18105:
18101:
18097:
18093:
18089:
18084:
18079:
18074:
18070:
18066:
18062:
18057:
18053:
18049:
18045:
18041:
18037:
18033:
18029:
18024:
18020:
18016:
18012:
18008:
18004:
18000:
17996:
17992:
17991:Pearson, Karl
17988:
17984:
17980:
17976:
17972:
17968:
17964:
17963:
17955:
17951:
17950:Pearson, Karl
17947:
17943:
17937:
17933:
17928:
17923:
17918:
17914:
17910:
17906:
17901:
17897:
17893:
17889:
17885:
17881:
17877:
17872:
17867:
17863:
17859:
17855:
17850:
17845:
17840:
17836:
17832:
17828:
17824:
17820:
17816:
17810:
17805:
17804:
17797:
17793:
17789:
17784:
17779:
17775:
17771:
17767:
17762:
17758:
17754:
17749:
17742:
17738:
17734:
17730:
17726:
17721:
17716:
17712:
17708:
17701:
17696:
17692:
17686:
17682:
17677:
17673:
17672:
17667:
17666:
17660:
17657:
17653:
17649:
17646:
17640:
17636:
17632:
17628:
17624:
17618:
17614:
17609:
17605:
17599:
17595:
17590:
17586:
17582:
17577:
17572:
17568:
17564:
17560:
17555:
17551:
17547:
17543:
17539:
17534:
17529:
17526:(1): 3:1â14.
17525:
17521:
17516:
17512:
17506:
17502:
17497:
17493:
17487:
17483:
17478:
17474:
17470:
17466:
17460:
17456:
17451:
17447:
17441:
17437:
17433:
17432:
17427:
17422:
17418:
17414:
17413:
17408:
17404:
17400:
17394:
17390:
17385:
17381:
17377:
17373:
17369:
17365:
17361:
17356:
17352:
17346:
17342:
17341:
17336:
17332:
17328:
17324:
17320:
17319:
17314:
17310:
17306:
17300:
17295:
17294:
17287:
17280:
17279:
17273:
17269:
17265:
17260:
17255:
17251:
17247:
17243:
17238:
17234:
17228:
17224:
17223:
17218:
17214:
17210:
17206:
17202:
17198:
17194:
17190:
17185:
17181:
17176:
17171:
17166:
17162:
17158:
17154:
17149:
17145:
17139:
17135:
17130:
17126:
17120:
17116:
17111:
17107:
17101:
17097:
17092:
17088:
17082:
17078:
17073:
17070:
17066:
17062:
17058:
17052:
17047:
17043:
17038:
17037:
17022:
17018:
17014:
17010:
17006:
17002:
16998:
16994:
16990:
16983:
16975:
16969:
16962:
16957:
16950:
16949:
16942:
16935:
16931:
16925:
16919:, p. 23)
16918:
16917:Maxwell (1860
16913:
16906:
16905:Stigler (1978
16901:
16894:
16893:Stigler (1978
16889:
16882:
16881:Stigler (1986
16877:
16870:
16869:Pearson (1905
16865:
16858:
16857:Laplace (1774
16853:
16846:
16841:
16834:
16829:
16823:, p. 76)
16822:
16821:Stigler (1986
16817:
16811:, p. 77)
16810:
16804:
16798:, p. 74)
16797:
16792:
16786:, p. 85)
16785:
16780:
16774:
16769:
16762:
16761:Monahan (1985
16757:
16751:
16750:Karney (2016)
16746:
16740:
16735:
16729:
16724:
16718:
16713:
16706:
16701:
16693:
16689:
16685:
16681:
16678:(3): 477â84.
16677:
16673:
16666:
16657:
16649:
16643:
16639:
16632:
16625:
16617:
16615:9780521592710
16611:
16607:
16606:
16598:
16592:
16591:Huxley (1932)
16587:
16580:
16575:
16568:
16563:
16556:
16551:
16549:
16540:
16536:
16532:
16528:
16521:
16514:(2): 257â263.
16513:
16509:
16505:
16498:
16492:
16487:
16485:
16477:
16472:
16465:
16460:
16452:
16446:
16442:
16435:
16427:
16423:
16419:
16415:
16412:(4): 359â62.
16411:
16407:
16406:
16398:
16390:
16386:
16382:
16378:
16373:
16368:
16364:
16360:
16359:
16354:
16347:
16339:
16335:
16328:
16322:, p. 27)
16321:
16316:
16308:
16304:
16299:
16294:
16290:
16286:
16281:
16276:
16272:
16268:
16264:
16257:
16255:
16239:
16238:Stat.ucla.edu
16235:
16229:
16214:
16210:
16204:
16198:
16193:
16182:
16175:
16159:
16155:
16149:
16142:
16141:
16134:
16127:
16126:
16119:
16113:, p. 35)
16112:
16107:
16105:
16097:
16096:0-340-52922-9
16093:
16089:
16083:
16075:
16069:
16061:
16055:
16051:
16046:
16045:
16036:
16028:
16022:
16018:
16013:
16012:
16003:
15996:
15991:
15985:, p. 24)
15984:
15979:
15973:, p. 23)
15972:
15967:
15958:
15953:
15946:
15938:
15931:
15924:
15919:
15912:
15907:
15900:
15895:
15893:
15891:
15882:
15878:
15874:
15870:
15866:
15862:
15858:
15854:
15850:
15846:
15842:
15838:
15834:
15830:
15829:
15824:
15818:
15809:
15790:
15786:
15782:
15777:
15772:
15768:
15764:
15757:
15750:
15742:
15740:9780471748816
15736:
15732:
15727:
15726:
15717:
15709:
15703:
15699:
15695:
15691:
15690:
15682:
15674:
15670:
15666:
15662:
15658:
15654:
15650:
15644:
15640:
15639:
15634:
15630:
15626:
15620:
15611:
15610:
15605:
15602:
15595:
15584:
15577:
15570:
15562:
15558:
15551:
15544:
15539:
15532:
15527:
15520:
15515:
15509:
15504:
15489:
15485:
15479:
15477:
15468:
15462:
15458:
15451:
15444:
15438:
15431:
15426:
15419:
15418:
15412:
15397:
15393:
15389:
15385:
15380:
15375:
15371:
15367:
15360:
15353:
15349:
15330:
15326:
15322:
15318:
15314:
15310:
15304:
15297:
15296:Pearson (1905
15293:
15287:
15280:
15274:
15268:
15267:Walker (1985)
15264:
15260:
15257:
15253:
15249:
15240:
15233:
15227:
15223:
15212:
15209:
15206:
15203:
15200:
15196:
15193:
15191:
15188:
15186:
15183:
15181:
15180:Stein's lemma
15178:
15176:
15173:
15171:
15168:
15166:
15163:
15161:
15158:
15155:
15138:
15134:
15131:
15121:
15118:
15115:
15104:
15098:
15095:
15090:
15087:
15083:
15074:
15068:
15058:
15051:
15045:
15042:
15039:
15008:
15001:
14997:
14992:
14987:
14984:
14978:
14965:
14962:
14959:
14954:
14950:
14946:
14943:
14937:
14934:
14929:
14926:
14923:
14919:
14912:
14909:
14904:
14900:
14894:
14888:
14882:
14856:
14853:
14842:
14839:
14836:
14832:
14831:Gaussian blur
14829:
14827:
14824:
14822:
14821:number theory
14818:
14815:
14812:
14809:
14806:
14803:
14800:
14797:
14796:
14792:
14786:
14781:
14771:
14756:
14752:
14748:
14741:
14737:
14733:
14704:
14701:
14698:
14670:
14665:
14661:
14655:
14651:
14646:
14640:
14637:
14634:
14631:
14628:
14623:
14619:
14613:
14610:
14607:
14604:
14601:
14596:
14592:
14585:
14580:
14574:
14571:
14568:
14565:
14562:
14557:
14553:
14547:
14544:
14541:
14538:
14535:
14530:
14526:
14519:
14509:
14503:
14500:
14497:
14494:
14491:
14486:
14482:
14476:
14473:
14470:
14467:
14464:
14459:
14455:
14448:
14443:
14437:
14434:
14431:
14428:
14425:
14420:
14416:
14410:
14407:
14404:
14401:
14398:
14393:
14389:
14382:
14372:
14366:
14363:
14360:
14357:
14354:
14349:
14345:
14339:
14336:
14333:
14330:
14327:
14322:
14318:
14311:
14306:
14300:
14297:
14294:
14290:
14285:
14281:
14279:
14273:
14270:
14267:
14260:
14257:
14234:
14231:
14228:
14207:
14201:
14198:
14194:
14190:
14187:
14184:
14180:
14158:
14155:
14151:
14127:
14124:
14116:
14113:
14110:
14106:
14102:
14097:
14090:
14071:
14067:
14064:
14058:
14055:
14052:
14049:
14046:
14043:
14040:
14034:
14030:
14024:
14018:
14015:
14012:
14009:
14006:
14000:
13996:
13990:
13984:
13981:
13978:
13972:
13968:
13962:
13957:
13952:
13948:
13942:
13939:
13935:
13928:
13922:
13919:
13914:
13911:
13906:
13900:
13886:
13883:
13880:
13876:
13873:
13870:
13866:
13858:
13855:
13849:
13842:
13835:
13828:
13821:
13818:= 0.2316419,
13814:
13810:
13806:
13790:
13784:
13779:
13775:
13771:
13768:
13764:
13759:
13756:
13752:
13746:
13740:
13737:
13733:
13727:
13723:
13717:
13713:
13709:
13704:
13700:
13694:
13690:
13686:
13681:
13677:
13671:
13667:
13663:
13658:
13654:
13648:
13644:
13640:
13637:
13632:
13628:
13623:
13616:
13610:
13607:
13604:
13601:
13595:
13581:
13573:
13569:
13562:
13558:
13554:
13551:
13550:
13540:
13536:
13533:
13529:
13525:
13521:
13516:
13515:
13511:
13507:
13503:
13499:
13496:
13492:
13488:
13484:
13481:Optional: if
13480:
13477:
13473:
13470:
13466:
13463:Optional: if
13462:
13459:
13455:
13449:
13441:
13437:
13434:
13430:
13426:
13425:
13423:
13420:
13416:
13397:
13393:
13390:
13387:
13384:
13381:
13374:
13371:
13368:
13364:
13358:
13354:
13351:
13348:
13345:
13342:
13335:
13332:
13329:
13321:
13316:
13312:
13308:
13303:
13299:
13295:
13291:
13288:
13284:
13280:
13276:
13271:
13267:
13262:
13258:
13254:
13250:
13234:
13228:
13225:
13222:
13216:
13213:
13207:
13204:
13201:
13198:
13195:
13190:
13187:
13183:
13177:
13174:
13171:
13165:
13162:
13156:
13153:
13150:
13147:
13144:
13139:
13136:
13129:
13125:
13121:
13117:
13113:
13109:
13105:
13102:
13098:
13094:
13090:
13087:
13084:
13080:
13076:
13072:
13068:
13064:
13061:property: if
13060:
13056:
13055:
13051:
13047:
13042:
13020:
13016:
13012:
13009:based on the
13008:
13004:
13000:
12996:
12992:
12989:
12985:
12981:
12977:
12973:
12969:
12965:
12961:
12960:
12957:
12952:
12945:
12941:
12937:
12933:
12929:
12926:
12924:in his works.
12923:
12919:
12915:
12911:
12907:
12903:
12899:
12895:
12891:
12886:
12883:
12879:
12875:
12871:
12868:
12867:
12865:
12861:
12857:
12853:
12852:
12848:
12838:
12834:
12829:
12817:
12816:BoseâEinstein
12813:
12810:
12805:
12802:
12799:
12796:
12795:
12793:
12789:
12785:
12784:
12775:
12771:
12767:
12748:
12742:
12719:
12716:
12713:
12707:
12699:
12695:
12685:
12673:
12670:
12665:
12659:
12656:
12653:
12647:
12641:
12624:
12620:
12616:
12612:
12608:
12604:
12601:
12597:
12596:
12592:
12587:
12575:
12572:
12568:
12565:
12561:
12558:
12557:
12543:
12539:
12535:
12531:
12527:
12523:
12519:
12515:
12511:
12507:
12504:
12500:
12497:
12493:
12492:
12463:
12459:
12455:
12454:harmonic mean
12436:
12433:
12430:
12425:
12422:
12411:
12410:harmonic mean
12407:
12403:
12399:
12395:
12391:
12375:
12370:
12367:
12357:
12354:
12350:
12346:
12341:
12338:
12334:
12327:
12319:
12316:
12311:
12306:
12303:
12297:
12292:
12286:
12283:
12280:
12275:
12272:
12262:
12259:
12255:
12251:
12232:
12229:
12226:
12221:
12218:
12215:
12212:
12209:
12198:
12197:
12183:
12179:
12175:
12171:
12168:
12165:
12161:
12157:
12154:
12150:
12147:
12143:
12140:
12139:
12130:
12126:
12123:
12121:
12118:
12117:
12113:
12109:
12106:
12104:
12101:
12099:
12096:
12095:
12072:
12065:
12052:
12046:
12038:
12031:
12024:
12016:
12009:
11999:
11990:
11985:
11981:
11978:
11975:
11971:
11967:
11963:
11960: â
11959:
11955:
11949:are equal to
11948:
11939:
11934:
11930:
11923:
11919:
11915:
11912:
11911:
11909:
11898:
11894:
11883:
11879:
11868:
11857:
11853:
11835:
11831:
11828:
11825:
11824:
11820:
11816:
11812:
11810:
11804:
11801:
11797:
11793:
11789:
11785:
11782:
11778:
11775:
11770:
11767:
11764:
11761:
11758:
11755:
11754:
11752:
11743:
11739:
11734:
11729:
11725:
11719:
11715:
11708:
11703:
11699:
11698:Hilbert space
11695:
11691:
11688:
11685:
11682:
11679:
11675:
11671:
11667:
11663:
11658:
11654:
11649:
11646:
11643:
11640:
11637:
11633:
11629:
11626:= 2 case are
11625:
11621:
11617:
11613:
11609:
11604:
11590:
11579:
11575:
11570:
11567:-dimensional
11566:
11562:
11558:
11557:
11533:
11517:
11514:
11509:
11505:
11501:
11498:
11493:
11488:
11484:
11480:
11477:
11474:
11466:
11462:
11461:
11441:
11436:
11433:
11430:
11426:
11422:
11416:
11412:
11407:
11401:
11396:
11392:
11388:
11385:
11382:
11377:
11372:
11368:
11364:
11359:
11354:
11350:
11345:
11339:
11335:
11330:
11324:
11319:
11315:
11311:
11308:
11305:
11300:
11295:
11291:
11287:
11282:
11277:
11273:
11268:
11261:
11258:
11248:
11244:
11237:
11218:
11214:
11210:
11207:
11204:
11199:
11195:
11191:
11186:
11182:
11159:
11155:
11151:
11148:
11145:
11140:
11136:
11132:
11127:
11123:
11114:
11100:
11095:
11092:
11089:
11085:
11081:
11074:
11068:
11055:
11050:
11045:
11041:
11034:
11031:
11028:
11023:
11010:
11005:
11000:
10996:
10988:
10978:
10975:
10972:
10966:
10962:
10955:
10952:
10944:
10940:
10936:
10933:
10930:
10925:
10921:
10912:
10909:
10901:
10893:
10887:
10883:
10878:
10875:
10867:
10859:
10856:
10836:
10833:
10830:
10822:
10818:
10814:
10810:
10806:
10803:, then their
10788:
10784:
10763:
10741:
10737:
10733:
10730:
10727:
10722:
10718:
10714:
10709:
10705:
10696:
10682:
10677:
10672:
10668:
10664:
10659:
10654:
10650:
10646:
10643:
10640:
10635:
10630:
10626:
10605:
10597:
10579:
10575:
10571:
10568:
10565:
10560:
10556:
10552:
10547:
10543:
10534:
10531:
10527:
10526:
10517:
10497:
10492:
10488:
10484:
10479:
10474:
10470:
10459:
10441:
10438:
10435:
10429:
10426:
10423:
10418:
10414:
10409:
10403:
10399:
10390:
10386:
10369:
10365:
10361:
10358:
10348:
10344:
10340:
10337:
10331:
10325:
10317:
10313:
10305:
10289:
10286:
10283:
10275:
10257:
10253:
10224:
10211:
10207:
10201:
10198:
10194:
10190:
10184:
10176:
10172:
10163:
10145:
10141:
10135:
10131:
10127:
10124:
10116:
10098:
10095:
10092:
10079:
10074:
10070:
10066:
10061:
10057:
10048:
10047:
10038:
10017:
10012:
10008:
10001:
9998:
9995:
9989:
9984:
9979:
9975:
9971:
9964:
9959:
9955:
9949:
9944:
9940:
9933:
9928:
9923:
9919:
9893:
9888:
9884:
9877:
9874:
9871:
9865:
9860:
9855:
9851:
9847:
9840:
9835:
9831:
9825:
9821:
9814:
9811:
9808:
9802:
9797:
9792:
9788:
9782:
9778:
9774:
9768:
9763:
9759:
9733:
9728:
9724:
9720:
9715:
9711:
9678:
9675:
9672:
9667:
9663:
9657:
9652:
9648:
9641:
9629:
9621:
9618:
9615:
9610:
9606:
9599:
9591:
9586:
9582:
9574:
9563:
9558:
9549:
9530:
9527:
9524:
9518:
9515:
9490:
9485:
9481:
9477:
9472:
9468:
9454:
9449:
9445:
9436:
9421:
9418:
9415:
9407:
9389:
9385:
9377:and variance
9364:
9344:
9341:
9333:
9329:
9325:
9320:
9316:
9310:
9304:
9301:
9298:
9292:
9287:
9283:
9279:
9276:
9271:
9267:
9263:
9257:
9252:
9248:
9227:
9207:
9185:
9181:
9173:and variance
9160:
9138:
9134:
9111:
9107:
9098:
9095:
9077:
9073:
9069:
9049:
9046:
9043:
9023:
9020:
9017:
8995:
8991:
8970:
8950:
8942:
8925:
8920:
8916:
8912:
8907:
8902:
8898:
8890:and variance
8875:
8871:
8867:
8862:
8858:
8835:
8831:
8827:
8822:
8818:
8795:
8790:
8786:
8763:
8758:
8754:
8731:
8727:
8704:
8700:
8691:
8673:
8669:
8646:
8642:
8633:
8632:
8609:
8606:
8602:
8593:
8575:
8572:
8564:
8561:
8558:
8548:
8545:
8535:{\textstyle }
8526:
8523:
8520:
8497:
8489:
8486:
8470:
8466:
8460:
8457:
8452:
8448:
8444:
8441:
8438:
8433:
8428:
8423:
8419:
8416:
8413:
8407:
8400:
8397:
8392:
8389:
8383:
8377:
8374:
8371:
8363:
8347:
8339:
8336:
8320:
8317:
8314:
8289:
8285:
8280:
8274:
8270:
8261:
8256:
8252:
8248:
8243:
8239:
8234:
8228:
8224:
8215:
8199:
8195:
8191:
8183:
8166:
8162:
8158:
8155:
8151:
8142:
8139:
8136:
8123:
8107:
8103:
8094:
8091:
8088:
8075:
8072:
8056:
8053:
8050:
8024:
8020:
8016:
8013:
8005:
8001:
7997:
7993:
7990:
7987:
7977:
7961:
7953:
7930:
7926:
7921:
7918:
7902:
7899:
7893:
7887:
7879:
7863:
7855:
7852:The standard
7851:
7828:
7824:
7820:
7817:
7811:
7805:
7802:
7799:
7794:
7790:
7781:
7765:
7757:
7754:
7736:
7732:
7726:
7722:
7714:and variance
7701:
7698:
7695:
7692:
7672:
7652:
7632:
7629:
7626:
7623:
7616:
7615:
7589:
7585:
7576:
7571:
7568:
7565:
7561:
7552:
7536:
7533:
7528:
7525:
7521:
7500:
7497:
7492:
7487:
7483:
7462:
7459:
7454:
7449:
7445:
7424:
7421:
7416:
7412:
7391:
7388:
7385:
7380:
7376:
7355:
7335:
7327:
7311:
7308:
7303:
7300:
7296:
7275:
7272:
7267:
7263:
7242:
7239:
7234:
7230:
7209:
7206:
7201:
7197:
7176:
7173:
7168:
7164:
7143:
7123:
7101:
7097:
7088:
7072:
7069:
7066:
7046:
7043:
7040:
7037:
7017:
6995:
6991:
6987:
6984:
6976:
6972:
6948:
6925:
6919:
6912:
6908:
6893:
6873:
6870:
6863:and variance
6850:
6827:
6819:
6815:
6807:
6803:
6788:
6768:
6761:and variance
6748:
6728:
6720:
6716:
6701:
6681:
6658:
6655:
6652:
6646:
6643:
6636:and variance
6623:
6620:
6612:
6593:
6590:
6587:
6581:
6574:
6570:
6569:
6567:
6547:
6544:
6524:
6501:
6495:
6486:
6478:
6446:
6443:
6440:
6437:
6432:
6429:
6421:
6417:
6411:
6400:
6375:
6371:
6367:
6364:
6358:
6336:
6332:
6328:
6325:
6322:
6317:
6313:
6304:
6284:
6251:
6235:
6219:
6216:
6209:
6205:
6201:
6197:
6194:
6190:
6186:
6182:
6178:
6163:
6157:
6154:
6149:
6141:
6136:
6132:
6128:
6123:
6116:
6112:
6108:
6102:
6097:
6089:
6084:
6080:
6076:
6071:
6066:
6062:
6048:
6040:
6035:
6031:
6027:
6022:
6018:
6012:
6007:
6003:
5993:
5982:
5977:
5973:
5969:
5966:
5963:
5958:
5954:
5950:
5947:
5927:
5902:
5897:
5893:
5889:
5884:
5880:
5873:
5870:
5867:
5842:
5838:
5834:
5831:
5825:
5822:
5800:
5796:
5792:
5789:
5786:
5781:
5777:
5768:
5764:
5748:
5737:
5733:
5729:
5725:
5718:
5711:
5702:
5698:
5694:
5686:
5681:
5673:
5669:
5665:
5662:
5630:
5626:
5605:
5597:
5593:
5578:
5569:
5564:
5560:
5556:
5551:
5546:
5542:
5534:
5524:
5520:
5516:
5511:
5507:
5495:
5492:
5487:
5483:
5479:
5476:
5467:
5462:
5458:
5454:
5449:
5444:
5440:
5432:
5428:
5422:
5418:
5414:
5407:
5404:
5401:
5393:
5389:
5385:
5380:
5376:
5367:
5363:
5354:
5337:
5329:
5324:
5320:
5314:
5309:
5305:
5299:
5296:
5293:
5290:
5287:
5280:
5275:
5271:
5265:
5260:
5256:
5249:
5243:
5240:
5235:
5227:
5222:
5218:
5214:
5207:
5197:
5193:
5189:
5184:
5180:
5170:
5162:
5158:
5154:
5149:
5145:
5127:
5119:is given by:
5101:
5096:
5092:
5088:
5083:
5079:
5072:
5069:
5064:
5060:
5052:from another
5034:
5029:
5025:
5021:
5016:
5012:
5005:
5002:
4997:
4993:
4984:
4980:
4977:
4961:
4941:
4933:
4929:
4925:
4909:
4889:
4881:
4878:
4862:
4842:
4834:
4831:(named after
4830:
4826:
4807:
4801:
4778:
4772:
4769:
4766:
4763:
4757:
4749:
4745:
4724:
4702:
4698:
4689:
4688:
4638:
4634:
4630:
4623:
4607:
4603:
4599:
4596:
4591:
4587:
4581:
4577:
4573:
4570:
4565:
4561:
4555:
4551:
4547:
4544:
4539:
4535:
4529:
4525:
4521:
4518:
4513:
4509:
4501:
4498:
4497:
4482:
4475:
4459:
4455:
4451:
4448:
4445:
4440:
4436:
4430:
4426:
4422:
4419:
4414:
4410:
4404:
4400:
4396:
4393:
4388:
4384:
4376:
4373:
4372:
4355:
4351:
4347:
4340:
4324:
4320:
4316:
4313:
4308:
4304:
4298:
4294:
4290:
4287:
4282:
4278:
4272:
4268:
4264:
4261:
4256:
4252:
4244:
4241:
4240:
4225:
4218:
4202:
4198:
4194:
4191:
4188:
4183:
4179:
4173:
4169:
4165:
4162:
4157:
4153:
4145:
4142:
4141:
4124:
4120:
4116:
4109:
4093:
4089:
4085:
4082:
4077:
4073:
4067:
4063:
4059:
4056:
4051:
4047:
4039:
4036:
4035:
4020:
4013:
3997:
3993:
3989:
3986:
3983:
3978:
3974:
3966:
3963:
3962:
3945:
3941:
3933:
3917:
3913:
3909:
3904:
3900:
3892:
3889:
3888:
3873:
3866:
3852:
3845:
3842:
3841:
3837:
3834:
3831:
3830:
3826:
3801:
3797:
3790:
3787:
3784:
3778:
3775:
3753:
3749:
3728:
3708:
3700:
3697:
3674:
3668:
3663:
3659:
3638:
3632:
3626:
3620:
3614:
3609:
3605:
3599:
3591:
3588:
3582:
3576:
3565:
3558:
3545:
3528:
3522:
3516:
3513:
3508:
3504:
3497:
3491:
3484:
3481:
3472:
3458:
3452:
3446:
3443:
3440:
3437:
3431:
3424:
3421:
3412:
3411:
3407:
3403:
3399:
3396:
3392:
3378:
3375:
3372:
3369:
3366:
3363:
3343:
3340:
3337:
3334:
3331:
3311:
3303:
3299:
3285:
3279:
3273:
3266:
3262:
3255:
3251:
3247:
3242:
3234:
3231:
3228:
3219:
3213:
3206:
3203:
3194:
3180:
3174:
3168:
3161:
3157:
3152:
3149:
3146:
3140:
3137:
3131:
3124:
3121:
3112:
3097:
3089:
3075:
3072:
3069:
3066:
3046:
3043:
3040:
3037:
3030:negative for
3017:
3014:
3011:
3008:
3000:
2996:
2992:
2989:
2985:
2981:
2965:
2962:
2959:
2956:
2948:
2947:
2920:
2917:
2903:
2900:
2899:
2884:
2881:
2867:
2864:
2863:
2848:
2845:
2831:
2828:
2827:
2812:
2809:
2795:
2792:
2791:
2776:
2773:
2759:
2756:
2755:
2740:
2737:
2723:
2720:
2719:
2704:
2701:
2687:
2684:
2683:
2666:
2662:
2654:
2640:
2633:
2614:
2610:
2602:
2588:
2581:
2580:
2576:
2555:
2544:
2543:
2540:
2526:
2512:
2509:
2508:
2495:
2484:
2483:
2480:
2466:
2452:
2449:
2448:
2435:
2427:
2426:
2423:
2409:
2395:
2392:
2391:
2387:
2383:
2379:
2364:
2359:
2358:
2355:
2341:
2327:
2324:
2323:
2319:
2315:
2311:
2296:
2291:
2290:
2287:
2273:
2259:
2256:
2255:
2251:
2247:
2243:
2225:
2220:
2219:
2216:
2202:
2188:
2185:
2184:
2181:
2178:
2161:
2158:
2155:
2144:
2132:
2118:
2115:
2112:
2105:
2088:
2085:
2082:
2079:
2073:
2070:
2064:
2061:
2058:
2055:
2049:
2046:
2043:
2036:
2022:
2015:
2014:
2006:
2001:
1997:
1968:
1964:
1935:
1931:
1907:
1873:
1869:
1859:
1854:
1850:
1846:
1843:
1830:
1829:
1776:
1771:
1769:
1764:
1762:
1757:
1756:
1754:
1753:
1748:
1745:
1743:
1740:
1739:
1738:
1737:
1732:
1729:
1727:
1724:
1722:
1719:
1717:
1714:
1712:
1709:
1707:
1704:
1703:
1702:
1701:
1696:
1693:
1691:
1688:
1686:
1683:
1681:
1678:
1677:
1676:
1675:
1668:
1665:
1663:
1660:
1658:
1655:
1653:
1650:
1648:
1645:
1643:
1640:
1638:
1635:
1633:
1630:
1629:
1628:
1625:
1623:
1620:
1616:
1613:
1611:
1608:
1606:
1603:
1601:
1598:
1596:
1593:
1591:
1588:
1587:
1586:
1583:
1579:
1576:
1575:
1574:
1571:
1567:
1564:
1562:
1559:
1557:
1554:
1552:
1549:
1547:
1544:
1543:
1542:
1539:
1537:
1534:
1532:
1529:
1528:
1527:
1526:
1521:
1518:
1516:
1515:Indeterminism
1513:
1509:
1506:
1505:
1504:
1501:
1497:
1494:
1493:
1492:
1489:
1488:
1487:
1486:
1482:
1478:
1477:
1474:
1471:
1470:
1467:
1463:
1462:
1440:
1437:
1434:
1426:
1420:
1415:
1410:
1405:
1401:
1398:
1395:
1389:
1383:
1379:
1374:
1369:
1363:
1356:
1353:
1349:
1341:
1338:
1335:
1326:
1322:
1306:
1298:
1293:
1289:
1283:
1278:
1274:
1268:
1265:
1262:
1259:
1256:
1249:
1244:
1240:
1233:
1223:
1219:
1215:
1210:
1206:
1196:
1191:
1186:
1179:
1175:
1169:
1165:
1159:
1153:
1147:
1144:
1133:
1129:
1112:
1101:
1097:
1093:
1086:
1082:
1077:
1070:
1063:
1059:
1054:
1050:
1044:
1039:
1031:
1027:
1023:
1020:
1001:
997:
979:
975:
969:
965:
959:
955:
951:
948:
945:
942:
936:
933:
924:
920:
902:
898:
892:
888:
882:
878:
874:
871:
868:
862:
859:
850:
846:
826:
822:
818:
815:
812:
806:
803:
798:
795:
784:
780:
765:
756:
752:
737:
728:
724:
707:
703:
699:
694:
685:
681:
664:
660:
650:
646:
631:
622:
618:
603:
594:
590:
575:
566:
562:
544:
541:
538:
535:
529:
524:
521:
517:
511:
506:
503:
500:
491:
487:
471:
466:
458:
453:
448:
445:
442:
436:
432:
429:
426:
423:
419:
413:
410:
405:
401:
396:
392:
389:
386:
380:
367:
363:
341:
337:
333:
326:
318:
315:
312:
303:
299:
290:
286:
282:
279:
275:
264:
260:
240:
237:
228:
224:
219:
215:
197:
194:
184:
179:
175:
165:
161:
140:
137:
128:
124:
104:
100:
96:
93:
72:
68:
60:
55:
49:
41:
33:
19:
19756:
19744:
19710:Multivariate
19709:
19697:
19685:
19680:Wrapped LĂ©vy
19640:
19588:Matrix gamma
19581:
19561:
19549:Normal-gamma
19542:
19508:Continuous:
19507:
19478:
19423:Tukey lambda
19410:
19402:
19397:-exponential
19394:
19386:
19377:
19368:
19359:
19353:-exponential
19350:
19294:
19267:
19261:
19228:
19190:
19177:
19104:Poly-Weibull
19049:Log-logistic
19009:
19008:Hotelling's
18940:
18782:Logit-normal
18656:GaussâKuzmin
18651:FloryâSchulz
18532:with finite
18461:
18420:
18415:
18403:the original
18398:
18349:
18345:
18319:
18293:
18272:
18245:
18241:
18214:
18210:
18178:
18174:
18149:
18145:
18120:
18116:
18091:
18087:
18068:
18064:
18038:(1): 25â45.
18035:
18031:
18002:
17998:
17966:
17960:
17931:
17912:
17908:
17887:
17886:. Series 4.
17883:
17861:
17857:
17834:
17830:
17802:
17773:
17769:
17756:
17752:
17741:the original
17710:
17706:
17680:
17669:
17664:
17647:
17644:
17638:
17612:
17593:
17566:
17562:
17523:
17519:
17500:
17481:
17454:
17430:
17410:
17388:
17366:(3): 12â14.
17363:
17359:
17339:
17322:
17317:
17292:
17277:
17249:
17245:
17221:
17192:
17179:
17160:
17156:
17133:
17114:
17095:
17076:
16996:
16992:
16982:
16968:
16956:
16946:
16941:
16929:
16924:
16912:
16900:
16888:
16876:
16864:
16852:
16840:
16828:
16816:
16809:Walker (1985
16803:
16791:
16779:
16768:
16763:, section 2)
16756:
16745:
16734:
16723:
16712:
16700:
16675:
16671:
16665:
16656:
16637:
16624:
16604:
16597:
16586:
16574:
16562:
16530:
16526:
16520:
16511:
16507:
16497:
16471:
16459:
16440:
16434:
16409:
16403:
16397:
16362:
16356:
16346:
16337:
16327:
16315:
16270:
16266:
16241:. Retrieved
16237:
16228:
16216:. Retrieved
16212:
16203:
16192:
16174:
16162:. Retrieved
16158:Allisons.org
16157:
16148:
16138:
16133:
16124:
16118:
16087:
16082:
16043:
16035:
16010:
16002:
15990:
15978:
15966:
15945:
15936:
15930:
15918:
15906:
15835:(1): 91â93.
15832:
15826:
15817:
15808:
15796:. Retrieved
15789:the original
15766:
15762:
15749:
15724:
15716:
15688:
15681:
15637:
15619:
15607:
15594:
15583:the original
15569:
15560:
15550:
15538:
15526:
15514:
15503:
15491:. Retrieved
15487:
15456:
15450:
15437:
15425:
15416:
15411:
15401:February 27,
15399:. Retrieved
15369:
15365:
15352:
15325:Lexis (1878)
15317:Galton (1889
15303:
15291:
15286:
15273:
15262:
15259:
15255:
15251:
15244:
15239:
15226:
15152:denotes the
14875:is given as
14095:
14088:
13847:
13840:
13833:
13826:
13819:
13812:
13808:
13804:
13571:
13567:
13560:
13556:
13531:
13509:
13508:then accept
13505:
13501:
13494:
13490:
13486:
13482:
13475:
13474:then accept
13471:
13468:
13464:
13457:
13453:
13447:
13439:
13432:
13428:
13418:
13414:
13319:
13314:
13310:
13306:
13301:
13297:
13286:
13282:
13269:
13265:
13260:
13256:
13127:
13123:
13118:distributed
13115:
13111:
13096:
13066:
13062:
13046:bean machine
12897:
12881:
12877:
12873:
12859:
12832:
12792:decomposable
12773:
12765:
12618:
12614:
12541:
12517:
12461:
12457:
12405:
12401:
12397:
12393:
12257:
12253:
12153:multivariate
12111:
11994:
11988:
11983:
11973:
11969:
11965:
11961:
11957:
11950:
11943:
11937:
11932:
11925:
11897:3-sigma rule
11808:
11747:
11741:
11737:
11727:
11723:
11717:
11713:
11706:
11701:
11677:
11676:matrix
11673:
11669:
11665:
11661:
11656:
11652:
11631:
11623:
11615:
11611:
11607:
11595:
11588:
11577:
11573:
11564:
11246:
11242:
10160:follows the
7780:log-normally
7550:
7325:
7086:
6974:
6886:, for large
4975:
4928:uncorrelated
4828:
2997:: its first
1747:Tree diagram
1742:Venn diagram
1706:Independence
1652:Markov chain
1604:
1536:Sample space
53:
19794:Exponential
19643:directional
19632:Directional
19519:Generalized
19490:Multinomial
19445:continuous-
19385:Kaniadakis
19376:Kaniadakis
19367:Kaniadakis
19358:Kaniadakis
19349:Kaniadakis
19301:TracyâWidom
19278:Skew normal
19260:Noncentral
19044:Log-Laplace
19022:Generalized
19003:Half-normal
18969:Generalized
18933:Logarithmic
18918:Exponential
18872:Chi-squared
18812:U-quadratic
18777:Kumaraswamy
18719:Continuous
18666:Logarithmic
18561:Categorical
17650:(3), 1986:
16845:Gauss (1809
16833:Gauss (1809
16728:Leva (1992)
16365:(1): 91â3.
15331:) c. 1875.
15279:Gauss (1809
14835:convolution
14753:proved the
14727:Development
13875:Cody (1969)
13865:West (2009)
13857:Hart (1968)
13578:(algorithm
13249:independent
13077:and in the
13075:Hart (1968)
12914:heavy tails
12770:convolution
12542:conditional
12469:Vector form
12390:reciprocals
12199:The factor
12194:Scalar form
11861:Sample mean
11807:Kaniadakis
11748:covariance
11571:. A vector
10805:sample mean
8690:independent
8485:chi-squared
8335:chi-squared
4932:independent
4827:, then the
3721:with known
3408:of order 2.
3406:supersmooth
3395:log-concave
2918:0.999999999
1786:Definitions
1662:Random walk
1503:Determinism
1491:Probability
19872:Categories
19789:Elliptical
19745:Degenerate
19731:Degenerate
19479:Discrete:
19438:univariate
19293:Student's
19248:Asymmetric
19227:Johnson's
19155:supported
19099:Phase-type
19054:Log-normal
19039:Log-Cauchy
19029:Kolmogorov
18947:Noncentral
18877:Noncentral
18857:Beta prime
18807:Triangular
18802:Reciprocal
18772:IrwinâHall
18721:univariate
18701:YuleâSimon
18583:Rademacher
18525:univariate
18065:Population
18032:Biometrika
17999:Biometrika
17641:: 621â656.
17457:. London.
17436:Free Press
17065:"Gaussian"
16320:Bryc (1995
16280:2012.14331
16111:Bryc (1995
15983:Bryc (1995
15971:Bryc (1995
15873:0060.28509
15561:Connexions
15493:August 15,
15379:1811.11301
15339:References
14115:Dia (2023)
13101:IrwinâHall
11891:See also:
11876:See also:
11865:See also:
11850:See also:
11788:q-Gaussian
11781:q-analogue
11636:ellipsoids
11554:Extensions
6674:for large
6613:with mean
4974:should be
4825:polynomial
3823:See also:
2999:derivative
2939:Properties
2882:0.99999999
1573:Experiment
1520:Randomness
1466:statistics
127:Parameters
19514:Dirichlet
19495:Dirichlet
19405:-Gaussian
19380:-Logistic
19217:Holtsmark
19189:Gaussian
19176:Fisher's
19159:real line
18661:Geometric
18641:Delaporte
18546:Bernoulli
18523:Discrete
18468:EMS Press
18383:MathWorld
18322:. Dover.
18195:122366147
18137:122148043
17983:125037489
17715:CiteSeerX
17533:1303.6257
17503:. Wiley.
17484:. Wiley.
17473:476909537
17417:EMS Press
17209:259689086
17098:. Wiley.
17021:237919587
17013:0361-0926
16418:0036-4452
16381:0003-4851
16338:MathWorld
16273:(10): 1.
16068:cite book
15957:1209.4340
15911:Fan (1991
15881:Q55897617
15849:0003-4851
15771:CiteSeerX
15609:MathWorld
15521:, item 7)
15396:254231768
15344:Citations
15088:α
15065:Ψ
15040:α
15034:Ψ
15002:β
14998:γ
14985:α
14974:Ψ
14963:γ
14947:β
14944:−
14938:
14927:−
14924:α
14910:α
14905:β
14860:∞
14656:−
14264:Φ
14261:−
14232:≥
14199:−
14191:×
14185:≈
14156:−
14131:Φ
14128:−
14068:⋯
14056:⋅
14050:⋅
14044:⋅
14016:⋅
14010:⋅
13982:⋅
13923:φ
13895:Φ
13741:ε
13611:φ
13608:−
13590:Φ
13391:
13382:−
13352:
13343:−
13226:π
13217:
13205:
13196:−
13175:π
13166:
13154:
13145:−
13120:uniformly
12995:hydrology
12898:logarithm
12860:logarithm
12692:∂
12682:∂
12639:∂
12635:∂
12607:diffusion
12536:used (an
12509:possible.
12368:−
12355:−
12339:−
12160:conjugate
12146:precision
12076:^
12073:σ
12056:^
12053:μ
12047:−
12000:), where
11534:variable.
11502:∑
11481:∑
11423:∼
11386:⋯
11309:⋯
11208:…
11149:…
11093:−
11082:∼
11059:¯
11051:−
11032:⋯
11014:¯
11006:−
10976:−
10956:μ
10953:−
10934:⋯
10879:μ
10876:−
10871:¯
10834:−
10785:σ
10764:μ
10731:…
10669:χ
10665:∼
10644:⋯
10569:…
10430:
10424:∼
10359:−
10314:ϕ
10199:−
10195:π
10080:∼
10067:±
10009:σ
10002:α
9999:−
9976:σ
9972:α
9956:σ
9941:σ
9924:α
9920:σ
9885:σ
9878:α
9875:−
9852:σ
9848:α
9832:σ
9815:α
9812:−
9789:σ
9775:α
9764:α
9729:α
9725:σ
9716:α
9679:α
9676:−
9658:α
9622:α
9619:−
9592:α
9564:∫
9519:∈
9482:σ
9455:∼
9416:α
9386:σ
9365:μ
9345:μ
9311:μ
9293:−
9182:σ
9161:μ
9074:σ
9047:−
8992:σ
8917:σ
8899:σ
8872:μ
8859:μ
8787:σ
8755:σ
8728:μ
8701:μ
8607:−
8603:σ
8573:−
8565:μ
8562:−
8487:variable.
8461:π
8453:σ
8445:
8439:−
8424:σ
8420:μ
8417:−
8393:−
8375:
8315:μ
8286:σ
8271:μ
8253:χ
8249:∼
8240:σ
8200:σ
8163:χ
8159:∼
8156:σ
8143:μ
8140:−
8108:σ
8095:μ
8092:−
8051:μ
8021:σ
8014:μ
7998:∼
7927:σ
7919:μ
7900:∼
7888:σ
7825:σ
7818:μ
7806:
7800:∼
7733:σ
7696:μ
7562:∑
7522:ρ
7484:σ
7446:σ
7413:μ
7389:−
7377:μ
7297:ρ
7264:σ
7231:σ
7198:μ
7165:μ
7067:σ
7044:−
7038:μ
6988:
6961:is large.
6949:ν
6926:ν
6816:χ
6789:λ
6769:λ
6749:λ
6729:λ
6656:−
6444:
6433:σ
6430:≤
6372:σ
6326:…
6278:∇
6245:∇
6217:−
6155:−
6133:σ
6113:σ
6081:σ
6063:σ
6052:¯
6032:σ
6019:μ
6004:σ
5983:∼
5967:…
5951:∣
5948:μ
5928:μ
5894:σ
5881:μ
5871:∼
5868:μ
5839:σ
5832:μ
5823:∼
5790:…
5734:σ
5699:σ
5670:σ
5663:μ
5627:σ
5606:μ
5561:σ
5543:σ
5521:μ
5517:−
5508:μ
5488:−
5480:
5459:σ
5441:σ
5429:σ
5419:σ
5408:−
5321:σ
5306:σ
5300:
5294:−
5288:−
5272:σ
5257:σ
5219:σ
5194:μ
5190:−
5181:μ
5155:∥
5093:σ
5080:μ
5070:∼
5026:σ
5013:μ
5003:∼
4877:cumulants
4770:
4746:ϕ
4699:ϕ
4635:σ
4604:σ
4588:σ
4578:μ
4562:σ
4552:μ
4536:σ
4526:μ
4510:μ
4456:σ
4452:μ
4437:σ
4427:μ
4411:σ
4401:μ
4385:μ
4352:σ
4321:σ
4305:σ
4295:μ
4279:σ
4269:μ
4253:μ
4199:σ
4195:μ
4180:σ
4170:μ
4154:μ
4121:σ
4090:σ
4074:σ
4064:μ
4048:μ
3994:σ
3990:μ
3975:μ
3942:σ
3914:σ
3901:μ
3853:μ
3802:σ
3791:μ
3788:−
3750:σ
3729:μ
3669:
3627:φ
3615:
3589:−
3559:φ
3523:φ
3514:−
3482:φ
3447:φ
3441:−
3422:φ
3404:, indeed
3376:σ
3370:μ
3344:σ
3341:−
3338:μ
3263:σ
3252:σ
3248:−
3235:μ
3232:−
3158:σ
3153:μ
3150:−
3141:−
3073:μ
3044:μ
3015:μ
2963:μ
2846:0.9999999
2159:−
2116:−
2089:σ
2083:−
2080:μ
2071:−
2065:σ
2056:μ
1958:Φ
1902:Φ
1863:Φ
1838:Φ
1566:Singleton
1438:−
1406:σ
1402:μ
1399:−
1370:−
1357:π
1342:σ
1339:−
1336:μ
1290:σ
1275:σ
1269:
1257:−
1241:σ
1220:μ
1216:−
1207:μ
1176:σ
1166:σ
1098:σ
1060:σ
1028:σ
1021:μ
956:σ
952:−
946:μ
937:
879:σ
869:μ
863:
823:σ
816:π
807:
708:π
695:σ
661:σ
632:μ
604:μ
576:μ
542:−
530:
522:−
507:σ
501:μ
454:σ
449:μ
446:−
433:
397:σ
393:μ
390:−
377:Φ
338:σ
319:μ
316:−
304:−
287:σ
283:π
241:∈
216:(squared
185:∈
176:σ
141:∈
138:μ
101:σ
94:μ
19847:Category
19779:Circular
19772:Families
19757:Singular
19736:singular
19500:Negative
19447:discrete
19413:-Weibull
19371:-Weibull
19255:Logistic
19139:Discrete
19109:Rayleigh
19089:Nakagami
19012:-squared
18986:Gompertz
18835:interval
18571:Negative
18556:Binomial
18368:18514848
18340:(1996).
18205:(1978).
18166:62021374
17993:(1905).
17952:(1901).
17737:15802663
17633:(1774).
17585:12884505
17550:14252035
17428:(1994).
17337:(1981).
17315:(1809).
17219:(1738).
16426:25048183
16307:34468706
16243:March 3,
16218:April 7,
16164:March 3,
15877:Wikidata
15673:65-12253
15657:64-60036
15026:, where
14777:See also
14767:â
14691:and for
13438:Compute
12976:z-scores
12972:stanines
12844:â
12164:improper
11980:PâP plot
11914:QâQ plot
11674:relation
11670:variance
11628:ellipses
10514:has the
8688:are two
8212:has the
7368:, where
7156:, where
6694:and for
5940:will be
5815:are iid
3485:″
3425:′
3207:″
3125:′
2995:unimodal
2986:and the
2810:0.999999
2141:or
1801:Notation
1647:Variance
727:Skewness
649:Variance
490:Quantile
214:variance
164:location
74:Notation
19857:Commons
19829:Wrapped
19824:Tweedie
19819:Pearson
19814:Mixture
19721:Bingham
19620:Complex
19610:Inverse
19600:Wishart
19593:Inverse
19580:Matrix
19554:Inverse
19470:(joint)
19389:-Erlang
19243:Laplace
19134:Weibull
18991:Shifted
18974:Inverse
18959:Fréchet
18882:Inverse
18817:Uniform
18737:Arcsine
18696:Skellam
18691:Poisson
18614:support
18588:Soliton
18541:Benford
18534:support
18470:, 2001
18262:2684031
18233:2958876
18108:2347972
18052:2331722
18019:2331536
17792:2236741
17656:2245476
17419:, 2001
17380:2681417
17268:2241949
17032:Sources
16692:2347330
16405:SankhyÄ
16389:2236166
16298:8419883
16052:, 366.
15865:0006626
15857:2236166
15798:June 2,
15665:0167642
14722:History
13580:26.2.17
13504:†â4 ln
13467:†5 â 4
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13444:√
13003:CumFreq
12980:t-tests
12940:IQ test
12856:biology
12768:is the
12158:Either
10272:is the
9010:, then
7854:sigmoid
6393:, then
6193:NEF-QVF
4976:jointly
3819:Moments
3690:is the
2774:0.99999
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2248::
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1894:is the
1561:Outcome
783:Entropy
227:Support
19763:Cantor
19605:Normal
19436:Mixed
19362:-Gamma
19288:Stable
19238:Landau
19212:Gumbel
19166:Cauchy
19094:Pareto
18906:Erlang
18887:Scaled
18842:Benini
18681:Panjer
18437:
18429:; and
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15737:
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15394:
15313:Galton
15309:Peirce
15292:normal
15211:Z-test
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13861:erfc()
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13565:|
13559:) for
12984:ANOVAs
12874:length
12858:, the
12814:has a
12625:
11972:) and
11922:rankit
11880:, and
11854:; and
11802:above.
11700:
10427:Cauchy
10245:where
9406:stable
9200:, and
8590:has a
8120:, has
7513:, and
7288:, and
3651:where
2993:It is
2984:median
2982:, the
2738:0.9999
1508:System
1496:Axioms
593:Median
19485:Ewens
19311:Voigt
19283:Slash
19064:Lomax
19059:Log-t
18964:Gamma
18911:Hyper
18901:Davis
18896:Dagum
18752:Bates
18742:ARGUS
18626:Borel
18425:, by
18406:(PDF)
18395:(PDF)
18364:S2CID
18316:(PDF)
18258:JSTOR
18229:JSTOR
18191:S2CID
18162:S2CID
18133:S2CID
18104:JSTOR
18048:JSTOR
18015:JSTOR
17979:S2CID
17965:. 6.
17957:(PDF)
17864:(4).
17837:(8).
17788:JSTOR
17744:(PDF)
17733:S2CID
17703:(PDF)
17668:[
17652:JSTOR
17581:S2CID
17546:S2CID
17528:arXiv
17376:JSTOR
17321:[
17282:(PDF)
17264:JSTOR
17205:S2CID
17017:S2CID
16934:Ch. 7
16688:JSTOR
16634:(PDF)
16422:JSTOR
16385:JSTOR
16275:arXiv
16184:(PDF)
15952:arXiv
15853:JSTOR
15792:(PDF)
15759:(PDF)
15586:(PDF)
15579:(PDF)
15392:S2CID
15374:arXiv
15362:(PDF)
15321:Lexis
15218:Notes
12878:inert
12566:; and
12549:Proof
12184:data.
11610:is a
11239:with
10823:with
10598:with
10035:(see
9751:with
8307:. If
8043:. If
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7030:with
6206:with
4823:is a
3832:Order
2922:6.109
2905:3.090
2901:0.998
2886:5.730
2869:2.807
2865:0.995
2850:5.326
2833:2.575
2814:4.891
2797:2.326
2778:4.417
2761:1.959
2742:3.890
2725:1.644
2706:3.290
2702:0.999
2689:1.281
2528:0.000
2514:0.999
2468:0.000
2454:0.999
2411:0.000
2397:0.999
2343:0.002
2329:0.997
2275:0.045
2261:0.954
2204:0.317
2190:0.682
1541:Event
218:scale
19734:and
19692:Kent
19119:Rice
19034:LĂ©vy
18862:Burr
18792:PERT
18757:Beta
18706:Zeta
18598:Zipf
18515:list
18435:ISBN
18324:ISBN
18298:ISBN
18279:ISBN
17936:ISBN
17809:ISBN
17685:ISBN
17617:ISBN
17598:ISBN
17505:ISBN
17486:ISBN
17469:OCLC
17459:ISBN
17440:ISBN
17393:ISBN
17345:ISBN
17299:ISBN
17227:ISBN
17193:SSRN
17138:ISBN
17119:ISBN
17100:ISBN
17081:ISBN
17009:ISSN
16642:ISBN
16610:ISBN
16445:ISBN
16414:ISSN
16377:ISSN
16303:PMID
16245:2017
16220:2024
16166:2017
16092:ISBN
16074:link
16054:ISBN
16021:ISBN
15845:ISSN
15800:2011
15735:ISBN
15702:ISBN
15669:LCCN
15653:LCCN
15643:ISBN
15495:2020
15461:ISBN
15403:2023
14702:<
14103:The
14098:= 10
13522:The
13431:and
13417:and
13300:and
13292:The
13126:and
13114:and
13106:The
13044:The
12982:and
12872:The
12790:and
12460:and
12404:and
12396:and
12256:and
11895:and
11832:The
11805:The
11786:the
11731:the
11634:are
11559:The
10528:Any
10037:here
9911:and
9036:and
8963:and
8661:and
7974:has
7665:and
7348:and
7136:and
7059:and
6909:The
6804:The
6717:The
6571:The
6269:and
6234:flat
5765:The
5618:and
5594:The
5351:The
4981:The
4954:and
4926:and
4922:are
4902:and
3741:and
3356:and
3041:>
3012:<
2988:mean
2980:mode
2829:0.99
2793:0.98
2757:0.95
2721:0.90
2685:0.80
2557:.897
2497:.893
2439:7673
2437:.192
2382:OEIS
2366:.398
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2303:5080
2298:.977
2246:OEIS
2227:.151
2180:OEIS
1998:and
1950:and
621:Mode
565:Mean
195:>
160:mean
19570:LKJ
18867:Chi
18354:doi
18250:doi
18219:doi
18183:doi
18154:doi
18125:doi
18096:doi
18073:doi
18040:doi
18007:doi
17971:doi
17917:doi
17892:doi
17866:doi
17839:doi
17778:doi
17725:doi
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17538:doi
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16285:doi
16050:209
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15869:Zbl
15837:doi
15781:doi
15767:150
15731:254
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15384:doi
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14935:exp
14188:1.1
13582:):
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13485:â„ 4
13214:sin
13163:cos
13079:erf
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2212:863
2209:507
2206:310
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2195:492
2192:689
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860:exp
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804:log
684:MAD
518:erf
430:erf
366:CDF
263:PDF
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3181:.
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91:(
86:N
56:.
34:.
20:)
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