5946:
1011:
68:
52:
15873:
7039:
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10083:
7470:. However, this tends to be complicated by the fact that this requires finding the roots of a high degree polynomial, and there can be multiple roots that represent local maxima. Also, while the maximum likelihood estimator is asymptotically efficient, it is relatively inefficient for small samples. The log-likelihood function for the Cauchy distribution for sample size
7250:, the sample mean will become increasingly variable as more observations are taken, because of the increased probability of encountering sample points with a large absolute value. In fact, the distribution of the sample mean will be equal to the distribution of the observations themselves; i.e., the sample mean of a large sample is no better (or worse) an estimator of
5126:
11577:
7695:
8110:
7034:{\displaystyle {\begin{aligned}\operatorname {E} &\propto \int _{-\infty }^{\infty }{\frac {x^{2}}{1+x^{2}}}\,dx=\int _{-\infty }^{\infty }1-{\frac {1}{1+x^{2}}}\,dx\\&=\int _{-\infty }^{\infty }dx-\int _{-\infty }^{\infty }{\frac {1}{1+x^{2}}}\,dx=\int _{-\infty }^{\infty }dx-\pi =\infty .\end{aligned}}}
9819:
5333:
2057:
7044:
By re-arranging the formula, one can see that the second moment is essentially the infinite integral of a constant (here 1). Higher even-powered raw moments will also evaluate to infinity. Odd-powered raw moments, however, are undefined, which is distinctly different from existing with the value of
12590:
5949:
A typical trajectory of sample means looks like long periods of slow convergence to zero, punctuated by large jumps away from zero, but never getting too far away. A typical trajectory of sample variances looks similar, but the jumps accumulate faster than the decay, diverging to
4663:
4883:
11833:
7916:
11367:
4822:
7112:
Because the parameters of the Cauchy distribution do not correspond to a mean and variance, attempting to estimate the parameters of the Cauchy distribution by using a sample mean and a sample variance will not succeed. For example, if an i.i.d. sample of size
2601:
13211:
the Cauchy distribution is applied to extreme events such as annual maximum one-day rainfalls and river discharges. The blue picture illustrates an example of fitting the Cauchy distribution to ranked monthly maximum one-day rainfalls showing also the 90%
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1038:
in 1824, with Cauchy only becoming associated with it during an academic controversy in 1853. Poisson noted that if the mean of observations following such a distribution were taken, the mean error did not converge to any finite number. As such,
11124:
7496:
5574:
13203:
The Cauchy distribution is often the distribution of observations for objects that are spinning. The classic reference for this is called the Gull's lighthouse problem and as in the above section as the Breit–Wigner distribution in particle
12269:
7922:
384:
10078:{\displaystyle f({\mathbf {x} };{\mathbf {\mu } },{\mathbf {\Sigma } },k)={\frac {\Gamma \left({\frac {1+k}{2}}\right)}{\Gamma ({\frac {1}{2}})\pi ^{\frac {k}{2}}\left|{\mathbf {\Sigma } }\right|^{\frac {1}{2}}\left^{\frac {1+k}{2}}}}.}
6331:
For the integral to exist (even as an infinite value), at least one of the terms in this sum should be finite, or both should be infinite and have the same sign. But in the case of the Cauchy distribution, both the terms in this sum
5161:
1928:
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13811:
11910:
12459:
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since the two halves of the integral both diverge and have opposite signs. The first raw moment is the mean, which, being odd, does not exist. (See also the discussion above about this.) This in turn means that all of the
2946:
1067:
If one stands in front of a line and kicks a ball with a direction (more precisely, an angle) uniformly at random towards the line, then the distribution of the point where the ball hits the line is a Cauchy distribution.
6344:) is undefined, and thus so is the mean. When the mean of a probability distribution function (PDF) is undefined, no one can compute a reliable average over the experimental data points, regardless of the sample's size.
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are undefined since they are all based on the mean. The variance—which is the second central moment—is likewise non-existent (despite the fact that the raw second moment exists with the value infinity).
5121:{\displaystyle \mathrm {KL} \left(p_{x_{0,1},\gamma _{1}}:p_{x_{0,2},\gamma _{2}}\right)=\log {\frac {\left(\gamma _{1}+\gamma _{2}\right)^{2}+\left(x_{0,1}-x_{0,2}\right)^{2}}{4\gamma _{1}\gamma _{2}}}.}
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is the height of the peak. The three-parameter
Lorentzian function indicated is not, in general, a probability density function, since it does not integrate to 1, except in the special case where
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13003:
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included it as an example in her 1748 calculus textbook. Despite its name, the first explicit analysis of the properties of the Cauchy distribution was published by the French mathematician
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11572:{\displaystyle \Pi _{\gamma }(dy)=\left(c_{1,\gamma }{\frac {1}{y^{1+\gamma }}}1_{\left\{y>0\right\}}+c_{2,\gamma }{\frac {1}{|y|^{1+\gamma }}}1_{\left\{y<0\right\}}\right)\,dy}
11144:
1843:
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8912:
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1921:
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465:
11923:
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is also standard Cauchy distributed. In particular, the average does not converge to the mean, and so the standard Cauchy distribution does not follow the law of large numbers.
884:
4255:
5717:
The Cauchy distribution is usually used as an illustrative counterexample in elementary probability courses, as a distribution with no well-defined (or "indefinite") moments.
13099:
7690:{\displaystyle {\hat {\ell }}(x_{1},\dotsc ,x_{n}\mid \!x_{0},\gamma )=-n\log(\gamma \pi )-\sum _{i=1}^{n}\log \left(1+\left({\frac {x_{i}-x_{0}}{\gamma }}\right)^{2}\right)}
7069:
6056:
6004:
1108:
927:
8105:{\displaystyle {\frac {d\ell }{d\gamma }}=\sum _{i=1}^{n}{\frac {2\left(x_{i}-x_{0}\right)^{2}}{\gamma (\gamma ^{2}+\left(x_{i}-x_{0}\right)^{2})}}-{\frac {n}{\gamma }}=0}
4672:
of the probability density. The original probability density may be expressed in terms of the characteristic function, essentially by using the inverse
Fourier transform:
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2821:
1759:
309:
11185:
8296:
8276:
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7325:
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1876:
1783:
1732:
1175:
589:
549:
513:
105:
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defined by restricting the standard Cauchy distribution to the interval . Such a truncated distribution has all moments (and the central limit theorem applies for
6103:
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3014:
4859:
1466:
1403:
1047:
with such a distribution was inappropriate, as it assumed a finite mean and variance. Despite this, Poisson did not regard the issue as important, in contrast to
8504:
5328:{\displaystyle {\begin{aligned}H(\gamma )&=-\int _{-\infty }^{\infty }f(x;x_{0},\gamma )\log(f(x;x_{0},\gamma ))\,dx\\&=\log(4\pi \gamma )\end{aligned}}}
2052:{\displaystyle f(x;\psi )={\frac {1}{\pi }}\,{\textrm {Im}}\left({\frac {1}{x-\psi }}\right)={\frac {1}{\pi }}\,{\textrm {Re}}\left({\frac {-i}{x-\psi }}\right)}
10997:
10977:
10957:
10937:
9811:
9621:
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1250:
1198:
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than any single observation from the sample. Similarly, calculating the sample variance will result in values that grow larger as more observations are taken.
6061:
Moments of sample lower than order 1 would converge to zero. Moments of sample higher than order 2 would diverge to infinity even faster than sample variance.
1230:
13183:
and where the maximum likelihood estimator is found using ordinary least squares showed the sampling distribution of the statistic is the Cauchy distribution.
13057:
in which all atoms interact in the same way with the frequency range contained in the line shape. Many mechanisms cause homogeneous broadening, most notably
1014:
Estimating the mean and standard deviation through a sample from a Cauchy distribution (bottom) does not converge as the size of the sample grows, as in the
3265:
1486:
10578:
10442:
10352:
14531:
12585:{\displaystyle {\tfrac {1}{X}}\sim \operatorname {Cauchy} \left({\tfrac {x_{0}}{x_{0}^{2}+\gamma ^{2}}},{\tfrac {\gamma }{x_{0}^{2}+\gamma ^{2}}}\right)}
11846:
6006:
looks like long periods of slow convergence to zero, punctuated by large jumps away from zero, but never getting too far away. A typical trajectory of
3610:
2867:
13957:
13523:
6427:
6354:
6058:
looks similar, but the jumps accumulate faster than the decay, diverging to infinity. These two kinds of trajectories are plotted in the figure.
1130:-axis) chosen uniformly (between -90° and +90°) at random. The intersection of the line with the x-axis is the Cauchy distribution with location
8564:
by maximum likelihood. The truncated sample mean using the middle 24% order statistics is about 88% as asymptotically efficient an estimator of
9789:
Like how the standard Cauchy distribution is the
Student t-distribution with one degree of freedom, the multidimensional Cauchy density is the
5135:
between two Cauchy distributions is symmetric and can be expressed as a function of the chi-squared divergence. Closed-form expression for the
10151:
14420:
13836:
12882:
3434:
8304:
6634:
4865:
at the origin: this corresponds to the fact that the Cauchy distribution does not have well-defined moments higher than the zeroth moment.
4467:
3604:
676:
13104:
6113:
5848:
14660:
14201:
8595:
is used to find the solution for the maximum likelihood estimate, the middle 24% order statistics can be used as an initial solution for
5842:, which also has the standard Cauchy distribution. Consequently, no matter how many terms we take, the sample average does not converge.
15886:
15143:
13033:
13015:
12777:
4658:{\displaystyle \varphi _{X}(t)=\operatorname {E} \left=\int _{-\infty }^{\infty }f(x;x_{0},\gamma )e^{ixt}\,dx=e^{ix_{0}t-\gamma |t|}.}
14283:
14055:
13739:
10294:
10236:
9343:
5348:
15051:
2219:
10725:
683:
15838:
13586:
Frederic, Chyzak; Nielsen, Frank (2019). "A closed-form formula for the
Kullback-Leibler divergence between Cauchy distributions".
12081:
7123:
5580:
2620:
12408:
11190:
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4174:
cannot be dropped. It is also an example of a more generalized version of the central limit theorem that is characteristic of all
67:
15704:
14916:
14675:
14524:
14296:
12672:
8956:
12615:
8510:
solution on a computer is typically required. The benefit of maximum likelihood estimation is asymptotic efficiency; estimating
1258:
15599:
15363:
14323:
13680:
11828:{\displaystyle \pi |x|=\operatorname {PV} \int _{\mathbb {R} \smallsetminus \lbrace 0\rbrace }(1-e^{ixy})\,{\frac {dy}{y^{2}}}}
5945:
10534:
9636:
8950:
should have a univariate Cauchy distribution. The characteristic function of a multivariate Cauchy distribution is given by:
7911:{\displaystyle {\frac {d\ell }{dx_{0}}}=\sum _{i=1}^{n}{\frac {2(x_{i}-x_{0})}{\gamma ^{2}+\left(x_{i}-\!x_{0}\right)^{2}}}=0}
5774:
156:
51:
15037:
13501:
13392:
13359:
3541:
12276:
11164:
3708:
2131:
15358:
15302:
15200:
14962:
14600:
13762:
13487:
17:
12945:
11362:{\displaystyle \operatorname {E} \left(e^{ixX}\right)=\exp \left(\int _{\mathbb {R} }(e^{ixy}-1)\Pi _{\gamma }(dy)\right)}
11005:
10786:
10687:
8686:
4817:{\displaystyle f(x;x_{0},\gamma )={\frac {1}{2\pi }}\int _{-\infty }^{\infty }\varphi _{X}(t;x_{0},\gamma )e^{-ixt}\,dt\!}
15644:
15378:
15231:
14906:
14650:
811:
15108:
12349:
12008:
10088:
The properties of multidimensional Cauchy distribution are then special cases of the multivariate
Student distribution.
8628:
6573:
4363:
15876:
15548:
15524:
15103:
14517:
9630:
We also can write this formula for complex variable. Then the probability density function of complex cauchy is :
211:
15912:
15745:
15622:
15583:
15555:
15529:
15447:
15373:
14796:
14544:
14262:
13423:
12020:
9790:
9624:
8810:
3927:
2964:
with one degree of freedom, and so it may be constructed by any method that constructs the
Student's t-distribution.
1018:(top). There can be arbitrarily large jumps in the estimates, as seen in the graphs on the bottom. (Click to expand)
15733:
15699:
15565:
15560:
15405:
15213:
14911:
14665:
14403:
2741:
2611:
2484:
2437:
302:
2596:{\displaystyle F(x;x_{0},\gamma )={\frac {1}{\pi }}\arctan \left({\frac {x-x_{0}}{\gamma }}\right)+{\frac {1}{2}}}
15483:
15396:
15368:
15277:
15226:
15098:
14881:
14846:
14439:
14019:(1978). "Maximum Likelihood Estimates of the Parameters of the Cauchy Distribution for Samples of Size 3 and 4".
11663:
11585:
8745:
4874:
4113:
2827:
13930:
Rothenberg, Thomas J.; Fisher, Franklin, M.; Tilanus, C.B. (1964). "A note on estimation from a Cauchy sample".
13857:
Cane, Gwenda J. (1974). "Linear
Estimation of Parameters of the Cauchy Distribution Based on Sample Quantiles".
12138:
9251:
3198:
15907:
15497:
15414:
15251:
15175:
14998:
14876:
14851:
14715:
14710:
14705:
14202:"Non-linear Integral Equations to Approximate Bivariate Densities with Given Marginals and Dependence Function"
8248:{\displaystyle \sum _{i=1}^{n}{\frac {\left(x_{i}-x_{0}\right)^{2}}{\gamma ^{2}+\left(x_{i}-x_{0}\right)^{2}}}}
5609:
3822:
3475:
761:
12732:
11129:
7417:
of the Cauchy distribution, the efficiency of the estimator decreases if more than 24% of the sample is used.
1816:
960:
of order greater than or equal to one; only fractional absolute moments exist. The Cauchy distribution has no
15927:
15813:
15679:
15387:
15236:
15168:
15153:
15046:
15020:
14952:
14791:
14685:
14680:
14622:
14607:
14468:
13225:
5140:
7413:
that is more efficient than using either the sample median or the full sample mean. However, because of the
1048:
15649:
15639:
15330:
15256:
14957:
14816:
13892:
Zhang, Jin (2010). "A Highly
Efficient L-estimator for the Location Parameter of the Cauchy Distribution".
12001:{\displaystyle \operatorname {Cauchy} (\mu ,\sigma )\sim {\textrm {t}}_{(\mathrm {df} =1)}(\mu ,\sigma )\,}
11147:
8882:
6074:
3146:
1477:
204:
15709:
10645:
9183:
4353:{\displaystyle \lim _{c\to \infty }{\frac {1}{c}}\int _{-c}^{c}x^{2}\rho (x)\,dx={\frac {2\gamma }{\pi }}}
3019:
1884:
626:
15694:
15689:
15634:
15570:
15514:
15335:
15322:
15113:
15058:
15010:
14801:
14730:
14595:
14463:
14458:
12723:
11912:
10912:{\displaystyle {\frac {aX+b}{cX+d}}\sim \operatorname {Cauchy} \left({\frac {a\psi +b}{c\psi +d}}\right)}
6518:
5728:
4189:
4167:. We see that there is no law of large numbers for any weighted sum of independent Cauchy distributions.
4066:
4019:
3973:
3881:
2957:
1022:
A function with the form of the density function of the Cauchy distribution was studied geometrically by
397:
14416:
Gull, S.F. (1988) Bayesian
Inductive Inference and Maximum Entropy. Kluwer Academic Publishers, Berlin.
13978:
Bloch, Daniel (1966). "A note on the estimation of the location parameters of the Cauchy distribution".
841:
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15604:
15423:
15205:
15158:
15027:
15003:
14983:
14826:
14700:
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13560:
7045:
infinity. The odd-powered raw moments are undefined because their values are essentially equivalent to
1762:
961:
941:
666:
567:
12726:, taking values on a circle, is derived from the Cauchy distribution by wrapping it around the circle.
11119:{\displaystyle {\frac {X-i}{X+i}}\sim \operatorname {CCauchy} \left({\frac {\psi -i}{\psi +i}}\right)}
8625:
The shape can be estimated using the median of absolute values, since for location 0 Cauchy variables
15833:
15617:
15578:
15452:
15289:
15133:
15078:
14976:
14940:
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14776:
13075:
13058:
7048:
6009:
5957:
1074:
893:
13384:
7104:
observations from it); yet for almost all practical purposes it behaves like a Cauchy distribution.
7085:
15519:
15307:
15073:
15032:
14947:
14901:
14841:
14806:
14695:
14590:
14540:
13441:
8807:
is said to have the multivariate Cauchy distribution if every linear combination of its components
6070:
3256:
13832:
13068:
Applications of the Cauchy distribution or its transformation can be found in fields working with
12264:{\displaystyle \tan \left(\pi \left(X-{\tfrac {1}{2}}\right)\right)\sim {\textrm {Cauchy}}(0,1)\,}
3457:
118:
15818:
15760:
15431:
15218:
15128:
15083:
15068:
14886:
14836:
14831:
14632:
14612:
14151:
14110:
Barnett, V. D. (1966). "Order
Statistics Estimators of the Location of the Cauchy Distribution".
12717:
8464:
6532:
4862:
3438:
379:{\displaystyle {\frac {1}{\pi }}\arctan \left({\frac {x-x_{0}}{\gamma }}\right)+{\frac {1}{2}}\!}
14988:
13415:
13409:
13062:
9151:
9082:
5569:{\displaystyle H(\gamma )=\int _{0}^{1}\log \,(Q'(p;\gamma ))\,\mathrm {d} p=\log(4\pi \gamma )}
3790:
2832:
1035:
15684:
15672:
15661:
15543:
15439:
15246:
14690:
14670:
14575:
14400:"CumFreq, free software for cumulative frequency analysis and probability distribution fitting"
13054:
11637:
7097:
6348:
3449:
2990:
2098:
14499:
14216:
13351:
13344:
9111:
9046:
8917:
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1741:
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15015:
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13243:
13217:
11170:
8281:
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7361:
7310:
4421:
4171:
3456:
coefficients. In addition, the family of Cauchy-distributed random variables is closed under
3445:
2970:
2065:
1848:
1794:
1768:
1717:
1160:
1044:
994:
with a probability density function that can be expressed analytically, the others being the
574:
526:
490:
149:
82:
75:
13376:
3603:
This can be proved by repeated integration with the PDF, or more conveniently, by using the
15858:
15853:
15848:
15843:
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15163:
14766:
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14720:
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14078:
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12595:
9145:
8598:
8567:
8540:
8513:
8408:
7703:
7426:
7389:
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7327:
are needed. One simple method is to take the median value of the sample as an estimator of
7283:
7253:
7226:
7199:
5457:
4235:
3406:
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1802:
1686:
1411:
1133:
1040:
1031:
976:
957:
15063:
8435:
6079:
3013:
positive-semidefinite covariance matrix with strictly positive diagonal entries, then for
1110:
in the x-y plane, and select a line passing the point, with its direction (angle with the
999:
8:
15922:
15792:
15317:
15297:
15267:
15241:
15123:
14935:
14871:
14435:
Tong Liu (2012), An intermediate distribution between Gaussian and Cauchy distributions.
13607:
Nielsen, Frank; Okamura, Kazuki (2023). "On f-Divergences Between Cauchy Distributions".
13485:
13377:
12609:
11160:
7355:
7077:
4838:
4175:
2797:
1786:
1443:
1380:
1363:
1015:
995:
991:
934:
619:
14485:
14246:
8506:. Therefore, whether solving for one parameter or for both parameters simultaneously, a
8486:
1180:
This definition gives a simple way to sample from the standard Cauchy distribution. Let
15823:
15312:
15093:
15088:
14993:
14930:
14925:
14781:
14771:
14655:
14368:"The Limiting Distribution of the Serial Correlation Coefficient in the Explosive Case"
14173:
14127:
14036:
14016:
13995:
13909:
13874:
13803:
13795:
13665:
13634:
13616:
13587:
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10962:
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10922:
9796:
9606:
9586:
9566:
9546:
9317:
8592:
8507:
7473:
7420:
6735:
The Cauchy distribution does not have finite moments of any order. Some of the higher
6311:
5586:
5144:
4449:
3126:
3106:
3086:
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1711:
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1183:
1113:
930:
754:
612:
600:
560:
476:
107:
14293:
3368:{\displaystyle \sum _{j=1}^{p}w_{j}{\frac {X_{j}}{Y_{j}}}\sim \mathrm {Cauchy} (0,1).}
1203:
15721:
15148:
14891:
14821:
14786:
14735:
14482:
14258:
13913:
13638:
13497:
13419:
13388:
13355:
13221:
13036:, while the Cauchy distribution is the (non-relativistic) Breit–Wigner distribution.
7088:, which implies that higher moments (or halves of moments) diverge if lower ones do.
6187:
5339:
4669:
3396:
2607:
1673:{\displaystyle f(x;x_{0},\gamma )={\frac {1}{\pi \gamma \left}}={1 \over \pi }\left,}
519:
390:
14314:
13807:
13703:
13472:
10630:{\displaystyle {\tfrac {1}{X}}\sim \operatorname {Cauchy} (0,{\tfrac {1}{\gamma }})}
10522:{\displaystyle X-Y\sim \operatorname {Cauchy} (x_{0}-x_{1},\gamma _{0}+\gamma _{1})}
10432:{\displaystyle X+Y\sim \operatorname {Cauchy} (x_{0}+x_{1},\gamma _{0}+\gamma _{1})}
8537:
using the sample median is only about 81% as asymptotically efficient as estimating
14896:
14570:
14509:
14379:
14250:
14163:
14123:
14119:
14070:
14032:
14028:
13991:
13987:
13943:
13939:
13901:
13870:
13866:
13785:
13777:
13695:
13661:
13626:
13532:
13460:
13025:
7383:
1023:
984:
980:
940:
The Cauchy distribution is often used in statistics as the canonical example of a "
13699:
4877:
between two Cauchy distributions has the following symmetric closed-form formula:
2826:
For the standard distribution, the cumulative distribution function simplifies to
14443:
14424:
14300:
14279:
13213:
13021:
11905:{\displaystyle \operatorname {Cauchy} (0,1)\sim {\textrm {t}}(\mathrm {df} =1)\,}
5771:
from the standard Cauchy distribution, then the sequence of their sample mean is
5460:
of a distribution can be defined in terms of its quantile density, specifically:
5136:
1881:
It is sometimes convenient to express the PDF in terms of the complex parameter
1735:
1366:
1027:
823:
807:
607:
141:
14476:
Earliest Uses: The entry on Cauchy distribution has some historical information.
13268:
7378:. Other, more precise and robust methods have been developed For example, the
6504:
zero, as can be seen by computing the integral. This again shows that the mean (
14969:
14352:
14074:
13790:
13284:
13279:
7379:
7073:
6514:
2941:{\displaystyle F(x;0,1)={\frac {1}{\pi }}\arctan \left(x\right)+{\frac {1}{2}}}
1790:
1370:
972:
945:
471:
14417:
14384:
14367:
14168:
13905:
13439:
Pillai N.; Meng, X.L. (2016). "An unexpected encounter with Cauchy and Lévy".
13264:
15901:
15592:
15340:
14627:
14233:
Lemons, Don S. (2002), "An Introduction to Stochastic Processes in Physics",
13630:
13247:
13050:
8740:
1798:
1010:
35:
13232:
13046:
7747:
by taking the first derivative produces the following system of equations:
7117:
is taken from a Cauchy distribution, one may calculate the sample mean as:
6739:
do exist and have a value of infinity, for example, the raw second moment:
5132:
2483:
The Cauchy distribution is the probability distribution with the following
1476:
The Cauchy distribution is the probability distribution with the following
31:
6285:{\displaystyle \int _{-\infty }^{a}xf(x)\,dx+\int _{a}^{\infty }xf(x)\,dx}
14504:
13652:
Vasicek, Oldrich (1976). "A Test for Normality Based on Sample Entropy".
13191:
Fitted cumulative Cauchy distribution to maximum one-day rainfalls using
3536:
3453:
968:
111:
14399:
13486:
Campbell B. Read; N. Balakrishnan; Brani Vidakovic; Samuel Kotz (2006).
10224:{\displaystyle kX+\ell \sim {\textrm {Cauchy}}(x_{0}k+\ell ,\gamma |k|)}
929:
with a uniformly distributed angle. It is also the distribution of the
14288:
14177:
14131:
14040:
13999:
13878:
13799:
13546:
13464:
13414:(1 ed.). Hoboken, New Jersey: John Wiley & Sons Inc. pp.
13072:. A 1958 paper by White derived the test statistic for estimators of
12935:{\displaystyle Y=\mu +X{\sqrt {Z}}\sim \operatorname {Cauchy} (\mu ,s)}
6736:
14436:
14254:
13187:
8395:{\displaystyle \min |x_{i}-x_{0}|\leq \gamma \leq \max |x_{i}-x_{0}|.}
6720:{\displaystyle \operatorname {E} =\gamma ^{p}\mathrm {sec} (\pi p/2).}
1440:
is a rotationally symmetric distribution on the plane, then the ratio
14490:
13681:"Maximum entropy autoregressive conditional heteroskedasticity model"
13346:
An Introduction to Probability Theory and Its Applications, Volume II
13239:
13208:
13065:
also gives rise to a line shape described by the Cauchy distribution.
13029:
10642:: Expressing a Cauchy distribution in terms of one complex parameter
7414:
3383:
The Cauchy distribution is an example of a distribution which has no
1738:
which specifies the half-width at half-maximum (HWHM), alternatively
815:
13781:
13537:
13518:
13176:{\displaystyle x_{t+1}=\beta {x}_{t}+\varepsilon _{t+1},\beta >1}
5935:{\displaystyle V_{n}={\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-S_{n})^{2}}
14475:
13621:
13592:
13455:
13312:
6190:
by computing the sum of two one-sided improper integrals. That is,
3388:
1374:
949:
595:
555:
13383:(3 ed.). Cambridge, UK: Cambridge University Press. pp.
14152:"A Representation of the Symmetric Bivariate Cauchy Distribution"
13192:
12872:{\displaystyle Z\sim \operatorname {Inverse-Gamma} (1/2,s^{2}/2)}
5342:, the quantile density function, for the Cauchy distribution is:
3698:{\displaystyle \varphi _{X}(t)=\operatorname {E} \left=e^{-|t|}.}
2213:
In physics, a three-parameter Lorentzian function is often used:
10342:{\displaystyle Y\sim \operatorname {Cauchy} (x_{1},\gamma _{1})}
10284:{\displaystyle X\sim \operatorname {Cauchy} (x_{0},\gamma _{0})}
9533:{\displaystyle f(x,y;x_{0},y_{0},\gamma )={1 \over 2\pi }\left.}
5446:{\displaystyle Q'(p;\gamma )=\gamma \,\pi \,{\sec }^{2}\left.\!}
3878:
are independent and Cauchy distributed with location parameters
12942:. For half-Cauchy distributions, the relation holds by setting
9543:
Note that in this example, even though the covariance between
9337:
An example of a bivariate Cauchy distribution can be given by:
9177:
a positive homogeneous function of degree one. More formally:
7101:
3400:
2404:{\displaystyle f(x;x_{0},\gamma ,I)={\frac {I}{\left}}=I\left,}
1714:, specifying the location of the peak of the distribution, and
483:
30:"Lorentz distribution" redirects here. Not to be confused with
14316:
Lévy processes and continuous-state branching processes:part I
10776:{\displaystyle X\sim \operatorname {Cauchy} (x_{0},|\gamma |)}
9813:
dimension Student distribution with one degree of freedom is:
4418:
converges in distribution to a Cauchy distribution with scale
1051:, who was to engage Cauchy in a long dispute over the matter.
743:{\displaystyle \displaystyle \exp(x_{0}\,i\,t-\gamma \,|t|)\!}
13375:
Riley, Ken F.; Hobson, Michael P.; Bence, Stephen J. (2006).
12126:{\displaystyle {\tfrac {X}{Y}}\sim {\textrm {Cauchy}}(0,1)\,}
7280:
Therefore, more robust means of estimating the central value
7186:{\displaystyle {\bar {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}}
3467:
2728:{\displaystyle Q(p;x_{0},\gamma )=x_{0}+\gamma \,\tan \left.}
14480:
13350:(2 ed.). New York: John Wiley & Sons Inc. pp.
12449:{\displaystyle X\sim \operatorname {Cauchy} (x_{0},\gamma )}
11231:{\displaystyle X\sim \operatorname {Stable} (\gamma ,0,0)\,}
10141:{\displaystyle X\sim \operatorname {Cauchy} (x_{0},\gamma )}
8879:
has a Cauchy distribution. That is, for any constant vector
6493:{\displaystyle \lim _{a\to \infty }\int _{-2a}^{a}xf(x)\,dx}
13250:) producing a much larger probability of extreme risk than
13235:, according to the Lorentz model, is a Cauchy distribution.
12706:{\displaystyle X\sim \operatorname {Cauchy} (\mu ,\gamma )}
9033:{\displaystyle \varphi _{X}(t)=e^{ix_{0}(t)-\gamma (t)},\!}
6417:{\displaystyle \lim _{a\to \infty }\int _{-a}^{a}xf(x)\,dx}
3384:
14097:
Introduction to Robust Estimation & Hypothesis Testing
12662:{\displaystyle X\sim {\textrm {Stable}}(1,0,\gamma ,\mu )}
4835:
th derivative of the characteristic function evaluated at
1312:{\displaystyle x=\tan \left(\pi (u-{\frac {1}{2}})\right)}
13009:
11715:
This last representation is a consequence of the formula
3535:
sample from the standard Cauchy distribution, then their
3448:
to which the Cauchy distribution belongs is closed under
2478:
10568:{\displaystyle X\sim \operatorname {Cauchy} (0,\gamma )}
9779:{\displaystyle f(z;z_{0},\gamma )={1 \over 2\pi }\left.}
5835:{\displaystyle S_{n}={\frac {1}{n}}\sum _{i=1}^{n}X_{i}}
4170:
This shows that the condition of finite variance in the
193:{\displaystyle \displaystyle x\in (-\infty ,+\infty )\!}
14056:"The Pitman estimator of the Cauchy location parameter"
13929:
10640:
McCullagh's parametrization of the Cauchy distributions
7700:
Maximizing the log likelihood function with respect to
6424:
which is zero. On the other hand, the related integral
3593:{\displaystyle {\bar {X}}={\frac {1}{n}}\sum _{i}X_{i}}
3462:
McCullagh's parametrization of the Cauchy distributions
12539:
12490:
12464:
12339:{\displaystyle X\sim \operatorname {Log-Cauchy} (0,1)}
12214:
12086:
10613:
10583:
5418:
4178:, of which the Cauchy distribution is a special case.
3780:{\displaystyle \varphi _{\sum _{i}X_{i}}(t)=e^{-n|t|}}
2701:
2203:{\displaystyle f(x;0,1)={\frac {1}{\pi (1+x^{2})}}.\!}
440:
13107:
13078:
12948:
12885:
12780:
12735:
12675:
12618:
12462:
12411:
12352:
12279:
12187:
12141:
12084:
12023:
11926:
11849:
11724:
11666:
11640:
11588:
11381:
11247:
11193:
11173:
11132:
11046:
11008:
10985:
10965:
10945:
10925:
10827:
10789:
10728:
10690:
10648:
10581:
10537:
10445:
10355:
10297:
10239:
10154:
10103:
9822:
9799:
9639:
9609:
9589:
9569:
9549:
9346:
9320:
9254:
9186:
9154:
9114:
9085:
9049:
8959:
8920:
8885:
8813:
8748:
8689:
8631:
8601:
8570:
8543:
8516:
8489:
8467:
8438:
8411:
8307:
8284:
8264:
8124:
7925:
7756:
7733:
7706:
7499:
7476:
7456:
7429:
7392:
7364:
7333:
7313:
7286:
7256:
7229:
7202:
7126:
7091:
7051:
6748:
6637:
6576:
6535:
6430:
6357:
6314:
6201:
6116:
6082:
6012:
5960:
5851:
5777:
5731:
5612:
5589:
5469:
5351:
5164:
4886:
4841:
4681:
4479:
4452:
4424:
4366:
4258:
4238:
4192:
4116:
4069:
4022:
3976:
3930:
3884:
3825:
3793:
3711:
3613:
3544:
3478:
3460:
with real coefficients. In this connection, see also
3409:
3268:
3201:
3149:
3129:
3109:
3089:
3069:
3022:
2993:
2973:
2870:
2835:
2806:
2744:
2623:
2496:
2440:
2420:
2222:
2134:
2101:
2068:
1931:
1887:
1851:
1819:
1801:
scale parameter, defining what would now be called a
1771:
1744:
1720:
1689:
1489:
1471:
1446:
1414:
1383:
1348:
1328:
1261:
1238:
1206:
1186:
1163:
1136:
1116:
1077:
956:
below). The Cauchy distribution does not have finite
896:
844:
764:
687:
686:
629:
577:
529:
493:
400:
312:
214:
160:
159:
121:
85:
14539:
13407:
12998:{\displaystyle X\sim {\textrm {N}}(0,1)I\{X\geq 0\}}
11033:{\displaystyle X\sim \operatorname {Cauchy} (\psi )}
10814:{\displaystyle X\sim \operatorname {Cauchy} (\psi )}
10715:{\displaystyle X\sim \operatorname {Cauchy} (\psi )}
8724:{\displaystyle \operatorname {median} (|X|)=\gamma }
6163:{\displaystyle \int _{-\infty }^{\infty }xf(x)\,dx.}
5155:
The entropy of the Cauchy distribution is given by:
1813:
The maximum value or amplitude of the Cauchy PDF is
57:
The purple curve is the standard Cauchy distribution
14199:
13438:
8734:
15918:Probability distributions with non-finite variance
14312:
13654:Journal of the Royal Statistical Society, Series B
13343:
13276:, the Fourier transform of the Cauchy distribution
13175:
13093:
12997:
12934:
12871:
12766:
12705:
12661:
12584:
12448:
12396:{\displaystyle \ln(X)\sim {\textrm {Cauchy}}(0,1)}
12395:
12338:
12263:
12173:
12125:
12070:
12000:
11904:
11827:
11704:
11652:
11626:
11571:
11361:
11230:
11179:
11138:
11118:
11032:
10991:
10971:
10951:
10931:
10911:
10813:
10775:
10714:
10676:
10629:
10567:
10521:
10431:
10341:
10283:
10223:
10140:
10077:
9805:
9778:
9615:
9595:
9575:
9555:
9532:
9326:
9303:
9239:
9169:
9136:
9100:
9071:
9032:
8942:
8906:
8871:
8799:
8723:
8676:{\displaystyle X\sim \mathrm {Cauchy} (0,\gamma )}
8675:
8614:
8583:
8556:
8529:
8498:
8475:
8453:
8424:
8394:
8290:
8270:
8247:
8104:
7910:
7739:
7719:
7689:
7482:
7462:
7442:
7405:
7370:
7346:
7319:
7299:
7269:
7242:
7215:
7185:
7063:
7033:
6719:
6621:{\displaystyle X\sim \mathrm {Cauchy} (0,\gamma )}
6620:
6562:
6492:
6416:
6320:
6284:
6162:
6097:
6050:
5998:
5934:
5834:
5763:
5701:
5595:
5568:
5445:
5327:
5120:
4861:. Observe that the characteristic function is not
4853:
4816:
4657:
4458:
4430:
4411:{\displaystyle {\frac {1}{n}}\sum _{i=1}^{n}X_{i}}
4410:
4352:
4244:
4224:
4159:
4102:
4055:
4008:
3962:
3916:
3870:
3808:
3779:
3697:
3592:
3523:
3422:
3367:
3247:
3187:
3135:
3115:
3095:
3075:
3055:
3005:
2979:
2940:
2853:
2815:
2788:
2738:It follows that the first and third quartiles are
2727:
2595:
2468:
2426:
2403:
2202:
2113:
2087:
2051:
1915:
1870:
1837:
1797:exploited such a density function in 1827 with an
1777:
1753:
1726:
1702:
1672:
1460:
1432:
1397:
1354:
1334:
1311:
1244:
1224:
1192:
1169:
1149:
1122:
1102:
921:
878:
789:
742:
654:
583:
543:
507:
459:
378:
290:
192:
133:
99:
13246:, Cauchy distributions can be used to model VAR (
13231:The expression for the imaginary part of complex
13049:, the Cauchy distribution describes the shape of
12716:The Cauchy distribution is a singular limit of a
9029:
8469:
7877:
7547:
5442:
4813:
4466:denote a Cauchy distributed random variable. The
2465:
2199:
738:
651:
540:
504:
375:
291:{\displaystyle {\frac {1}{\pi \gamma \,\left}}\!}
287:
188:
96:
15899:
14275:
14273:
13524:Proceedings of the American Mathematical Society
13379:Mathematical Methods for Physics and Engineering
13374:
13313:N. L. Johnson; S. Kotz; N. Balakrishnan (1994).
12594:The Cauchy distribution is a limiting case of a
8353:
8308:
6432:
6359:
4260:
3607:of the standard Cauchy distribution (see below):
15933:Location-scale family probability distributions
14500:GNU Scientific Library – Reference Manual
14200:Molenberghs, Geert; Lesaffre, Emmanuel (1997).
14112:Journal of the American Statistical Association
14021:Journal of the American Statistical Association
13980:Journal of the American Statistical Association
13932:Journal of the American Statistical Association
13859:Journal of the American Statistical Association
13585:
12601:The Cauchy distribution is a special case of a
12071:{\displaystyle X,Y\sim {\textrm {N}}(0,1)\,X,Y}
8872:{\displaystyle Y=a_{1}X_{1}+\cdots +a_{k}X_{k}}
3963:{\displaystyle \gamma _{1},\ldots ,\gamma _{n}}
14505:Ratios of Normal Variables by George Marsaglia
13606:
13039:
6338:) are infinite and have opposite sign. Hence (
4868:
14525:
14270:
14241:(2), The Johns Hopkins University Press: 35,
14063:Journal of Statistical Planning and Inference
13925:
13923:
13833:"Illustration of instability of sample means"
13315:Continuous Univariate Distributions, Volume 1
9793:with one degree of freedom. The density of a
7223:will be concentrated about the central value
3435:infinitely divisible probability distribution
2789:{\displaystyle (x_{0}-\gamma ,x_{0}+\gamma )}
2469:{\displaystyle I={\frac {1}{\pi \gamma }}.\!}
1200:be a sample from a uniform distribution from
13956:: CS1 maint: multiple names: authors list (
12992:
12980:
11769:
11763:
10091:
7423:can also be used to estimate the parameters
6521:, fail to hold for the Cauchy distribution.
6513:Various results in probability theory about
1252:from the standard Cauchy distribution using
14418:https://doi.org/10.1007/978-94-009-3049-0_4
14365:
14053:
14011:
14009:
11705:{\displaystyle c_{1,\gamma }=c_{2,\gamma }}
11627:{\displaystyle c_{1,\gamma },c_{2,\gamma }}
11167:of such a stable distribution of parameter
8800:{\displaystyle X=(X_{1},\ldots ,X_{k})^{T}}
7107:
7084:The results for higher moments follow from
6105:, then the mean, if it exists, is given by
4160:{\displaystyle \sum _{i}|a_{i}|\gamma _{i}}
1059:Here are the most important constructions.
14532:
14518:
13920:
13408:Balakrishnan, N.; Nevrozov, V. B. (2003).
13368:
12174:{\displaystyle X\sim {\textrm {U}}(0,1)\,}
9304:{\displaystyle \gamma (at)=|a|\gamma (t),}
6351:of the mean of the Cauchy distribution is
4441:
3468:Sum of Cauchy-distributed random variables
3248:{\displaystyle w_{i}\geq 0,i=1,\ldots ,p,}
14383:
14292:, volume 79 (1992), pages 247–259.
14284:"Conditional inference and Cauchy models"
14167:
13789:
13729:
13620:
13591:
13536:
13454:
12260:
12170:
12122:
12058:
11997:
11901:
11802:
11756:
11634:can be expressed explicitly. In the case
11562:
11298:
11227:
8894:
8591:as the maximum likelihood estimate. When
8468:
6978:
6891:
6832:
6483:
6407:
6275:
6235:
6150:
5702:{\displaystyle \operatorname {E} =\log 4}
5533:
5503:
5385:
5381:
5283:
4806:
4600:
4325:
3871:{\displaystyle X_{1},X_{2},\ldots ,X_{n}}
3524:{\displaystyle X_{1},X_{2},\ldots ,X_{n}}
2674:
2010:
1963:
790:{\displaystyle {\frac {1}{2\gamma ^{2}}}}
721:
711:
707:
417:
227:
14346:
14195:
14193:
14149:
14145:
14143:
14141:
14015:
14006:
13973:
13971:
13969:
13967:
13678:
13186:
12767:{\displaystyle X\sim {\textrm {N}}(0,1)}
11838:
11139:{\displaystyle \operatorname {CCauchy} }
8483:requires solving a polynomial of degree
8432:requires solving a polynomial of degree
5944:
5581:maximum entropy probability distribution
4181:
2956:The standard Cauchy distribution is the
1838:{\displaystyle {\frac {1}{\pi \gamma }}}
1009:
14392:
14109:
13651:
13609:IEEE Transactions on Information Theory
13519:"A characterization of the Cauchy type"
13238:As an additional distribution to model
13220:. The rainfall data are represented by
11002:Using the same convention as above, if
4470:of the Cauchy distribution is given by
3403:are well defined and are both equal to
3015:independent and identically distributed
14:
15900:
14232:
14094:
13760:
13725:
13723:
13516:
13341:
13308:
13306:
13304:
13034:relativistic Breit–Wigner distribution
13016:Relativistic Breit–Wigner distribution
13010:Relativistic Breit–Wigner distribution
2951:
2479:Cumulative distribution function (CDF)
2125:with the probability density function
1468:has the standard Cauchy distribution.
1405:has the standard Cauchy distribution.
1062:
814:. It is also known, especially among
14513:
14481:
14372:The Annals of Mathematical Statistics
14190:
14156:The Annals of Mathematical Statistics
14138:
13977:
13964:
13891:
13679:Park, Sung Y.; Bera, Anil K. (2009).
13561:"Updates to the Cauchy Central Limit"
13411:A Primer on Statistical Distributions
13401:
13335:
13332:, S M Stigler Harvard 1999 Chapter 18
8907:{\displaystyle a\in \mathbb {R} ^{k}}
3188:{\displaystyle w_{1}+\cdots +w_{p}=1}
15882:
13856:
13489:Encyclopedia of Statistical Sciences
11660:of the Cauchy distribution, one has
10677:{\displaystyle \psi =x_{0}+i\gamma }
9240:{\displaystyle x_{0}(at)=ax_{0}(t),}
6192:
6107:
4063:is Cauchy distributed with location
3816:has a standard Cauchy distribution.
3056:{\displaystyle X,Y\sim N(0,\Sigma )}
1916:{\displaystyle \psi =x_{0}+i\gamma }
1808:
1026:in 1659, and later was known as the
655:{\displaystyle \log(4\pi \gamma )\!}
14437:https://arxiv.org/pdf/1208.5109.pdf
13720:
13301:
5764:{\displaystyle X_{1},X_{2},\ldots }
4831:th moment of a distribution is the
4225:{\displaystyle X_{1},X_{2},\ldots }
4103:{\displaystyle \sum _{i}a_{i}x_{i}}
4056:{\displaystyle \sum _{i}a_{i}X_{i}}
4009:{\displaystyle a_{1},\ldots ,a_{n}}
3917:{\displaystyle x_{1},\ldots ,x_{n}}
3444:Like all stable distributions, the
1071:More formally, consider a point at
812:continuous probability distribution
460:{\displaystyle x_{0}+\gamma \,\tan}
24:
14054:Cohen Freue, Gabriella V. (2007).
13666:10.1111/j.2517-6161.1976.tb01566.x
13328:Cauchy and the Witch of Agnesi in
12824:
12821:
12818:
12815:
12812:
12806:
12803:
12800:
12797:
12794:
12791:
12788:
12314:
12311:
12308:
12305:
12302:
12299:
12293:
12290:
12287:
11967:
11964:
11888:
11885:
11383:
11333:
11248:
9902:
9871:
8654:
8651:
8648:
8645:
8642:
8639:
7092:Moments of truncated distributions
7058:
7052:
7021:
7001:
6996:
6948:
6943:
6921:
6916:
6855:
6850:
6795:
6790:
6753:
6690:
6687:
6684:
6638:
6599:
6596:
6593:
6590:
6587:
6584:
6524:
6442:
6369:
6255:
6210:
6130:
6125:
5613:
5535:
5204:
5199:
4891:
4888:
4744:
4739:
4548:
4543:
4502:
4270:
3636:
3343:
3340:
3337:
3334:
3331:
3328:
3047:
2974:
1472:Probability density function (PDF)
879:{\displaystyle f(x;x_{0},\gamma )}
182:
173:
25:
15944:
14451:
13432:
9791:multivariate Student distribution
6730:
5720:
3458:linear fractional transformations
1232:, then we can generate a sample,
937:random variables with mean zero.
890:-intercept of a ray issuing from
15881:
15872:
15871:
14406:from the original on 2018-02-21.
13742:from the original on 9 July 2021
10027:
10008:
9980:
9943:
9851:
9831:
8735:Multivariate Cauchy distribution
7382:of the middle 24% of the sample
2614:) of the Cauchy distribution is
2485:cumulative distribution function
1054:
66:
64:Cumulative distribution function
50:
14429:
14410:
14359:
14340:
14329:from the original on 2016-03-03
14306:
14226:
14103:
14088:
14047:
13885:
13850:
13839:from the original on 2017-03-24
13825:
13814:from the original on 2022-01-25
13754:
13672:
13645:
13600:
13579:
13094:{\displaystyle {\hat {\beta }}}
11159:The Cauchy distribution is the
11154:
7064:{\displaystyle \infty -\infty }
6186:We may evaluate this two-sided
6051:{\displaystyle V_{1},V_{2},...}
5999:{\displaystyle S_{1},S_{2},...}
5845:Similarly, the sample variance
5579:The Cauchy distribution is the
1103:{\displaystyle (x_{0},\gamma )}
971:, it is closely related to the
922:{\displaystyle (x_{0},\gamma )}
14124:10.1080/01621459.1966.10482205
14033:10.1080/01621459.1978.10480031
13992:10.1080/01621459.1966.10480912
13944:10.1080/01621459.1964.10482170
13871:10.1080/01621459.1974.10480163
13770:Canadian Journal of Statistics
13763:"Is statistics too difficult?"
13553:
13510:
13479:
13322:
13085:
13063:Lifetime or natural broadening
12974:
12962:
12929:
12917:
12866:
12831:
12761:
12749:
12700:
12688:
12656:
12632:
12443:
12424:
12390:
12378:
12365:
12359:
12333:
12321:
12257:
12245:
12167:
12155:
12119:
12107:
12055:
12043:
11994:
11982:
11977:
11960:
11945:
11933:
11898:
11881:
11868:
11856:
11799:
11774:
11737:
11729:
11512:
11503:
11401:
11392:
11351:
11342:
11329:
11304:
11224:
11206:
11165:Lévy–Khintchine representation
11027:
11021:
10808:
10802:
10770:
10766:
10758:
10741:
10709:
10703:
10624:
10603:
10562:
10550:
10516:
10464:
10426:
10374:
10336:
10310:
10278:
10252:
10218:
10214:
10206:
10177:
10135:
10116:
10042:
10022:
9996:
9975:
9918:
9905:
9862:
9826:
9749:
9725:
9703:
9699:
9668:
9643:
9503:
9480:
9460:
9448:
9428:
9425:
9394:
9350:
9295:
9289:
9282:
9274:
9267:
9258:
9231:
9225:
9206:
9197:
9164:
9158:
9131:
9125:
9095:
9089:
9066:
9060:
9021:
9015:
9006:
9000:
8976:
8970:
8788:
8755:
8712:
8708:
8700:
8696:
8670:
8658:
8385:
8357:
8340:
8312:
8077:
8021:
7840:
7814:
7591:
7582:
7564:
7512:
7506:
7133:
6772:
6759:
6711:
6694:
6667:
6657:
6648:
6644:
6615:
6603:
6557:
6542:
6480:
6474:
6439:
6404:
6398:
6366:
6272:
6266:
6232:
6226:
6147:
6141:
6092:
6086:
5923:
5896:
5684:
5681:
5657:
5637:
5628:
5619:
5563:
5551:
5530:
5527:
5515:
5504:
5479:
5473:
5372:
5360:
5318:
5306:
5280:
5277:
5252:
5246:
5237:
5212:
5178:
5172:
4784:
4759:
4710:
4685:
4646:
4638:
4581:
4556:
4496:
4490:
4322:
4316:
4267:
4143:
4128:
3800:
3771:
3763:
3745:
3739:
3686:
3678:
3630:
3624:
3551:
3433:The Cauchy distribution is an
3359:
3347:
3050:
3038:
2892:
2874:
2848:
2842:
2783:
2745:
2652:
2627:
2525:
2500:
2369:
2349:
2257:
2226:
2190:
2171:
2156:
2138:
1947:
1935:
1638:
1618:
1518:
1493:
1427:
1415:
1301:
1282:
1219:
1207:
1097:
1078:
944:" distribution since both its
916:
897:
873:
848:
735:
731:
723:
694:
648:
636:
454:
451:
430:
424:
185:
167:
13:
1:
14366:White, J.S. (December 1958).
13700:10.1016/j.jeconom.2008.12.014
13295:
13226:cumulative frequency analysis
12608:The Cauchy distribution is a
6308:for an arbitrary real number
4360:is finite, but nonzero, then
3378:
14150:Ferguson, Thomas S. (1962).
11148:circular Cauchy distribution
4232:are and IID sample with PDF
2123:standard Cauchy distribution
1789:and is sometimes called the
1478:probability density function
953:
134:{\displaystyle \gamma >0}
48:Probability density function
7:
14464:Encyclopedia of Mathematics
14313:Kyprianou, Andreas (2009).
14235:American Journal of Physics
13258:
13040:Occurrence and applications
12724:wrapped Cauchy distribution
12009:non-standardized Student's
8476:{\displaystyle \,\!\gamma }
7196:Although the sample values
6563:{\displaystyle p\in (-1,1)}
6519:strong law of large numbers
6506:
6340:
6334:
6298:
6176:
4875:Kullback–Leibler divergence
4869:Kullback–Leibler divergence
886:is the distribution of the
838:. The Cauchy distribution
828:Cauchy–Lorentz distribution
10:
15949:
15705:Wrapped asymmetric Laplace
14676:Extended negative binomial
14303:from McCullagh's homepage.
14075:10.1016/j.jspi.2006.05.002
13517:Knight, Franck B. (1976).
13028:, the energy profile of a
13013:
9170:{\displaystyle \gamma (t)}
9101:{\displaystyle \gamma (t)}
8258:is a monotone function in
7307:and the scaling parameter
5712:
5150:
3809:{\displaystyle {\bar {X}}}
2854:{\displaystyle \arctan(x)}
1785:is also equal to half the
1763:full width at half maximum
1005:
962:moment generating function
29:
15867:
15801:
15759:
15660:
15496:
15474:
15465:
15364:Generalized extreme value
15349:
15184:
15144:Relativistic Breit–Wigner
14860:
14757:
14748:
14641:
14561:
14552:
14541:Probability distributions
14215:: 713–738. Archived from
13906:10.1007/s00180-009-0163-y
11653:{\displaystyle \gamma =1}
10092:Transformation properties
9625:statistically independent
7386:produces an estimate for
6529:The absolute moments for
5725:If we take an IID sample
5141:Jensen–Shannon divergence
3006:{\displaystyle p\times p}
2114:{\displaystyle \gamma =1}
836:Breit–Wigner distribution
758:
753:
680:
675:
670:
665:
623:
618:
611:
606:
599:
594:
571:
566:
559:
554:
523:
518:
487:
482:
475:
470:
394:
389:
306:
301:
208:
203:
153:
148:
79:
74:
62:
46:
15913:Continuous distributions
13894:Computational Statistics
13694:(2). Elsevier: 219–230.
13631:10.1109/TIT.2022.3231645
13442:The Annals of Statistics
13342:Feller, William (1971).
9137:{\displaystyle x_{0}(t)}
9108:are real functions with
9072:{\displaystyle x_{0}(t)}
8943:{\displaystyle Y=a^{T}X}
7108:Estimation of parameters
6071:probability distribution
5954:A typical trajectory of
5942:also does not converge.
3437:. It is also a strictly
3257:categorical distribution
2816:{\displaystyle 2\gamma }
1754:{\displaystyle 2\gamma }
27:Probability distribution
15359:Generalized chi-squared
15303:Normal-inverse Gaussian
14385:10.1214/aoms/1177706450
14169:10.1214/aoms/1177704357
13688:Journal of Econometrics
13330:Statistics on the Table
13233:electrical permittivity
12718:hyperbolic distribution
11180:{\displaystyle \gamma }
8461:, and solving just for
8291:{\displaystyle \gamma }
8271:{\displaystyle \gamma }
7740:{\displaystyle \gamma }
7463:{\displaystyle \gamma }
7371:{\displaystyle \gamma }
7320:{\displaystyle \gamma }
6064:
4468:characteristic function
4442:Characteristic function
4431:{\displaystyle \gamma }
4016:are real numbers, then
3605:characteristic function
2980:{\displaystyle \Sigma }
2088:{\displaystyle x_{0}=0}
1871:{\displaystyle x=x_{0}}
1778:{\displaystyle \gamma }
1727:{\displaystyle \gamma }
1170:{\displaystyle \gamma }
952:are undefined (but see
584:{\displaystyle \gamma }
544:{\displaystyle x_{0}\!}
508:{\displaystyle x_{0}\!}
100:{\displaystyle x_{0}\!}
15671:Univariate (circular)
15232:Generalized hyperbolic
14661:Conway–Maxwell–Poisson
14651:Beta negative binomial
13761:Hampel, Frank (1998),
13199:
13177:
13095:
13055:homogeneous broadening
12999:
12936:
12873:
12768:
12707:
12663:
12586:
12450:
12397:
12340:
12265:
12175:
12127:
12072:
12002:
11906:
11829:
11706:
11654:
11628:
11573:
11363:
11232:
11181:
11140:
11120:
11034:
10993:
10973:
10953:
10933:
10913:
10815:
10777:
10716:
10678:
10631:
10569:
10523:
10433:
10349:are independent, then
10343:
10285:
10225:
10142:
10079:
9807:
9780:
9617:
9597:
9577:
9557:
9534:
9328:
9305:
9241:
9171:
9138:
9102:
9073:
9034:
8944:
8914:, the random variable
8908:
8873:
8801:
8725:
8677:
8616:
8585:
8558:
8531:
8500:
8477:
8455:
8426:
8396:
8292:
8278:and that the solution
8272:
8249:
8145:
8106:
7969:
7912:
7807:
7741:
7721:
7691:
7617:
7484:
7464:
7444:
7407:
7372:
7348:
7321:
7301:
7271:
7244:
7217:
7187:
7172:
7098:truncated distribution
7065:
7035:
6721:
6622:
6564:
6494:
6418:
6349:Cauchy principal value
6322:
6286:
6164:
6099:
6052:
6000:
5951:
5936:
5895:
5836:
5821:
5765:
5703:
5597:
5570:
5447:
5338:The derivative of the
5329:
5147:, etc. are available.
5122:
4855:
4818:
4659:
4460:
4432:
4412:
4397:
4354:
4246:
4226:
4161:
4104:
4057:
4010:
3964:
3918:
3872:
3810:
3781:
3699:
3594:
3525:
3450:linear transformations
3424:
3369:
3289:
3249:
3189:
3137:
3117:
3097:
3077:
3057:
3007:
2981:
2942:
2855:
2817:
2790:
2729:
2597:
2470:
2428:
2405:
2204:
2115:
2089:
2062:The special case when
2053:
1917:
1872:
1839:
1779:
1755:
1728:
1704:
1674:
1462:
1434:
1399:
1356:
1336:
1313:
1246:
1226:
1194:
1171:
1151:
1124:
1104:
1019:
923:
880:
791:
744:
656:
585:
545:
509:
461:
380:
292:
194:
135:
101:
15908:Augustin-Louis Cauchy
15716:Bivariate (spherical)
15214:Kaniadakis κ-Gaussian
14486:"Cauchy Distribution"
14459:"Cauchy distribution"
14095:Wilcox, Rand (2012).
13732:"Cauchy Distribution"
13494:John Wiley & Sons
13252:Gaussian Distribution
13244:computational finance
13218:binomial distribution
13190:
13178:
13096:
13053:which are subject to
13000:
12937:
12874:
12769:
12708:
12664:
12587:
12451:
12398:
12341:
12266:
12176:
12128:
12073:
12003:
11907:
11839:Related distributions
11830:
11707:
11655:
11629:
11574:
11364:
11233:
11182:
11141:
11121:
11035:
10994:
10974:
10954:
10934:
10914:
10816:
10778:
10717:
10679:
10632:
10570:
10524:
10434:
10344:
10286:
10226:
10143:
10080:
9808:
9781:
9618:
9598:
9578:
9558:
9535:
9329:
9306:
9242:
9172:
9139:
9103:
9074:
9035:
8945:
8909:
8874:
8802:
8731:the shape parameter.
8726:
8678:
8617:
8615:{\displaystyle x_{0}}
8586:
8584:{\displaystyle x_{0}}
8559:
8557:{\displaystyle x_{0}}
8532:
8530:{\displaystyle x_{0}}
8501:
8478:
8456:
8427:
8425:{\displaystyle x_{0}}
8397:
8293:
8273:
8250:
8125:
8107:
7949:
7913:
7787:
7742:
7722:
7720:{\displaystyle x_{0}}
7692:
7597:
7485:
7465:
7445:
7443:{\displaystyle x_{0}}
7408:
7406:{\displaystyle x_{0}}
7373:
7349:
7347:{\displaystyle x_{0}}
7322:
7302:
7300:{\displaystyle x_{0}}
7272:
7270:{\displaystyle x_{0}}
7245:
7243:{\displaystyle x_{0}}
7218:
7216:{\displaystyle x_{i}}
7188:
7152:
7066:
7036:
6722:
6623:
6565:
6495:
6419:
6323:
6287:
6165:
6100:
6053:
6001:
5948:
5937:
5875:
5837:
5801:
5766:
5704:
5598:
5583:for a random variate
5571:
5448:
5330:
5123:
4856:
4819:
4660:
4461:
4433:
4413:
4377:
4355:
4247:
4245:{\displaystyle \rho }
4227:
4182:Central limit theorem
4172:central limit theorem
4162:
4105:
4058:
4011:
3965:
3919:
3873:
3811:
3782:
3700:
3595:
3526:
3446:location-scale family
3425:
3423:{\displaystyle x_{0}}
3370:
3269:
3250:
3190:
3138:
3118:
3098:
3078:
3058:
3008:
2982:
2943:
2856:
2818:
2791:
2730:
2598:
2471:
2429:
2406:
2205:
2116:
2090:
2054:
1918:
1873:
1840:
1795:Augustin-Louis Cauchy
1780:
1756:
1729:
1705:
1703:{\displaystyle x_{0}}
1675:
1463:
1435:
1433:{\displaystyle (U,V)}
1400:
1357:
1337:
1314:
1247:
1227:
1195:
1172:
1152:
1150:{\displaystyle x_{0}}
1125:
1105:
1045:central limit theorem
1013:
990:It is one of the few
924:
881:
832:Lorentz(ian) function
792:
745:
657:
586:
546:
510:
462:
381:
293:
195:
136:
102:
15928:Stable distributions
15781:Dirac delta function
15728:Bivariate (toroidal)
15685:Univariate von Mises
15556:Multivariate Laplace
15448:Shifted log-logistic
14797:Continuous Bernoulli
13274:Laplace distribution
13197:distribution fitting
13105:
13076:
13059:collision broadening
13032:is described by the
12946:
12883:
12778:
12733:
12673:
12616:
12603:Pearson distribution
12596:Pearson distribution
12460:
12409:
12350:
12277:
12185:
12139:
12082:
12021:
11924:
11847:
11722:
11664:
11638:
11586:
11379:
11245:
11191:
11171:
11130:
11044:
11006:
10983:
10963:
10943:
10923:
10825:
10787:
10726:
10688:
10646:
10579:
10535:
10443:
10353:
10295:
10237:
10152:
10101:
9820:
9797:
9637:
9607:
9587:
9567:
9547:
9344:
9318:
9252:
9184:
9152:
9146:homogeneous function
9112:
9083:
9047:
8957:
8918:
8883:
8811:
8746:
8687:
8629:
8599:
8568:
8541:
8514:
8487:
8465:
8454:{\displaystyle 2n-1}
8436:
8409:
8305:
8282:
8262:
8122:
7923:
7754:
7731:
7704:
7497:
7474:
7454:
7427:
7390:
7362:
7354:and half the sample
7331:
7311:
7284:
7254:
7227:
7200:
7124:
7078:standardized moments
7049:
6746:
6635:
6574:
6533:
6428:
6355:
6312:
6199:
6114:
6098:{\displaystyle f(x)}
6080:
6010:
5958:
5849:
5775:
5729:
5610:
5587:
5467:
5458:differential entropy
5349:
5162:
4884:
4839:
4679:
4477:
4450:
4422:
4364:
4256:
4236:
4190:
4176:stable distributions
4114:
4067:
4020:
3974:
3928:
3882:
3823:
3791:
3709:
3611:
3542:
3476:
3407:
3266:
3199:
3147:
3127:
3107:
3087:
3067:
3020:
2991:
2971:
2868:
2833:
2804:
2742:
2621:
2494:
2438:
2418:
2220:
2132:
2099:
2066:
1929:
1885:
1849:
1817:
1803:Dirac delta function
1769:
1742:
1718:
1687:
1487:
1444:
1412:
1381:
1364:normally distributed
1362:are two independent
1346:
1326:
1259:
1236:
1204:
1184:
1161:
1134:
1114:
1075:
992:stable distributions
977:fundamental solution
935:normally distributed
894:
842:
820:Lorentz distribution
762:
684:
627:
575:
527:
491:
398:
310:
212:
157:
119:
83:
18:Lorentz distribution
15829:Natural exponential
15734:Bivariate von Mises
15700:Wrapped exponential
15566:Multivariate stable
15561:Multivariate normal
14882:Benktander 2nd kind
14877:Benktander 1st kind
14666:Discrete phase-type
14247:2003AmJPh..71..191L
14017:Ferguson, Thomas S.
13791:20.500.11850/145503
12610:stable distribution
12559:
12517:
11161:stable distribution
7358:as an estimator of
7356:interquartile range
7086:Hölder's inequality
7005:
6952:
6925:
6859:
6799:
6467:
6391:
6259:
6219:
6134:
5499:
5208:
4854:{\displaystyle t=0}
4748:
4552:
4302:
3819:More generally, if
3705:With this, we have
2952:Other constructions
2828:arctangent function
2798:interquartile range
1787:interquartile range
1461:{\displaystyle U/V}
1408:More generally, if
1398:{\displaystyle U/V}
1063:Rotational symmetry
1016:normal distribution
996:normal distribution
933:of two independent
804:Cauchy distribution
43:
15484:Rectified Gaussian
15369:Generalized Pareto
15227:Generalized normal
15099:Matrix-exponential
14483:Weisstein, Eric W.
14442:2020-06-24 at the
14423:2022-01-25 at the
14299:2010-06-10 at the
14118:(316): 1205–1218.
13567:. 13 November 2022
13465:10.1214/15-AOS1407
13317:. New York: Wiley.
13290:Slash distribution
13222:plotting positions
13200:
13173:
13091:
13070:exponential growth
12995:
12932:
12869:
12764:
12703:
12659:
12582:
12575:
12545:
12533:
12503:
12473:
12446:
12393:
12336:
12261:
12223:
12171:
12123:
12095:
12078:independent, then
12068:
11998:
11902:
11825:
11702:
11650:
11624:
11569:
11359:
11228:
11177:
11136:
11116:
11030:
10989:
10969:
10949:
10929:
10909:
10811:
10773:
10712:
10674:
10627:
10622:
10592:
10565:
10519:
10429:
10339:
10281:
10221:
10138:
10075:
9803:
9776:
9613:
9593:
9573:
9553:
9530:
9324:
9301:
9237:
9167:
9148:of degree one and
9134:
9098:
9069:
9030:
8940:
8904:
8869:
8797:
8721:
8673:
8612:
8581:
8554:
8527:
8499:{\displaystyle 2n}
8496:
8473:
8451:
8422:
8392:
8288:
8268:
8245:
8102:
7908:
7737:
7717:
7687:
7480:
7460:
7440:
7421:Maximum likelihood
7403:
7368:
7344:
7317:
7297:
7267:
7240:
7213:
7183:
7061:
7031:
7029:
6988:
6935:
6908:
6842:
6782:
6717:
6618:
6560:
6490:
6447:
6446:
6414:
6374:
6373:
6318:
6282:
6245:
6202:
6160:
6117:
6095:
6048:
5996:
5952:
5932:
5832:
5761:
5699:
5593:
5566:
5485:
5443:
5427:
5325:
5323:
5191:
5145:Hellinger distance
5118:
4851:
4814:
4731:
4668:which is just the
4655:
4535:
4456:
4428:
4408:
4350:
4285:
4274:
4242:
4222:
4157:
4126:
4100:
4079:
4053:
4032:
4006:
3960:
3914:
3868:
3806:
3777:
3726:
3695:
3590:
3579:
3521:
3420:
3365:
3245:
3185:
3133:
3113:
3093:
3073:
3053:
3003:
2977:
2938:
2851:
2813:
2786:
2725:
2710:
2593:
2466:
2424:
2401:
2200:
2111:
2085:
2049:
1913:
1868:
1835:
1775:
1751:
1724:
1712:location parameter
1700:
1670:
1458:
1430:
1395:
1377:1, then the ratio
1352:
1332:
1309:
1242:
1222:
1190:
1167:
1147:
1120:
1100:
1020:
919:
876:
787:
755:Fisher information
740:
739:
652:
581:
541:
505:
457:
449:
376:
288:
190:
189:
131:
97:
41:
15895:
15894:
15492:
15491:
15461:
15460:
15352:whose type varies
15298:Normal (Gaussian)
15252:Hyperbolic secant
15201:Exponential power
15104:Maxwell–Boltzmann
14852:Wigner semicircle
14744:
14743:
14716:Parabolic fractal
14706:Negative binomial
14347:E. Hecht (1987).
14255:10.1119/1.1526134
14209:Statistica Sinica
13503:978-0-471-15044-2
13394:978-0-511-16842-0
13361:978-0-471-25709-7
13101:for the equation
13088:
12959:
12906:
12811:
12746:
12629:
12574:
12532:
12472:
12375:
12298:
12242:
12222:
12152:
12104:
12094:
12040:
11956:
11878:
11823:
11529:
11449:
11110:
11071:
10999:are real numbers.
10992:{\displaystyle d}
10972:{\displaystyle c}
10952:{\displaystyle b}
10932:{\displaystyle a}
10903:
10858:
10621:
10591:
10174:
10070:
10066:
9960:
9933:
9916:
9894:
9806:{\displaystyle k}
9767:
9687:
9616:{\displaystyle y}
9596:{\displaystyle x}
9576:{\displaystyle y}
9556:{\displaystyle x}
9521:
9413:
9327:{\displaystyle t}
8405:Solving just for
8243:
8094:
8081:
7944:
7900:
7782:
7670:
7509:
7483:{\displaystyle n}
7150:
7136:
6976:
6889:
6830:
6570:are defined. For
6431:
6358:
6321:{\displaystyle a}
6306:
6305:
6188:improper integral
6184:
6183:
5873:
5799:
5596:{\displaystyle X}
5426:
5340:quantile function
5113:
4729:
4670:Fourier transform
4459:{\displaystyle X}
4375:
4348:
4283:
4259:
4117:
4070:
4023:
3803:
3717:
3570:
3568:
3554:
3322:
3136:{\displaystyle Y}
3116:{\displaystyle X}
3096:{\displaystyle w}
3076:{\displaystyle p}
2936:
2906:
2709:
2608:quantile function
2591:
2574:
2539:
2460:
2427:{\displaystyle I}
2392:
2324:
2307:
2194:
2043:
2015:
2008:
1991:
1968:
1961:
1833:
1809:Properties of PDF
1661:
1606:
1593:
1575:
1355:{\displaystyle V}
1335:{\displaystyle U}
1299:
1245:{\displaystyle x}
1193:{\displaystyle u}
1123:{\displaystyle x}
1000:Lévy distribution
800:
799:
785:
448:
373:
356:
321:
285:
267:
58:
16:(Redirected from
15940:
15885:
15884:
15875:
15874:
15814:Compound Poisson
15789:
15777:
15746:von Mises–Fisher
15742:
15730:
15718:
15680:Circular uniform
15676:
15596:
15540:
15511:
15472:
15471:
15374:Marchenko–Pastur
15237:Geometric stable
15154:Truncated normal
15047:Inverse Gaussian
14953:Hyperexponential
14792:Beta rectangular
14760:bounded interval
14755:
14754:
14623:Discrete uniform
14608:Poisson binomial
14559:
14558:
14534:
14527:
14520:
14511:
14510:
14496:
14495:
14472:
14446:
14433:
14427:
14414:
14408:
14407:
14396:
14390:
14389:
14387:
14378:(4): 1188–1197.
14363:
14357:
14356:
14351:(2nd ed.).
14344:
14338:
14337:
14335:
14334:
14328:
14321:
14310:
14304:
14277:
14268:
14267:
14230:
14224:
14223:
14221:
14206:
14197:
14188:
14187:
14185:
14184:
14171:
14162:(4): 1256–1266.
14147:
14136:
14135:
14107:
14101:
14100:
14092:
14086:
14085:
14083:
14077:. Archived from
14060:
14051:
14045:
14044:
14027:(361): 211–213.
14013:
14004:
14003:
13986:(316): 852–855.
13975:
13962:
13961:
13955:
13947:
13938:(306): 460–463.
13927:
13918:
13917:
13889:
13883:
13882:
13865:(345): 243–245.
13854:
13848:
13847:
13845:
13844:
13829:
13823:
13821:
13820:
13819:
13793:
13767:
13758:
13752:
13751:
13749:
13747:
13727:
13718:
13717:
13715:
13714:
13708:
13702:. Archived from
13685:
13676:
13670:
13669:
13649:
13643:
13642:
13624:
13615:(5): 3150–3171.
13604:
13598:
13597:
13595:
13583:
13577:
13576:
13574:
13572:
13565:Quantum Calculus
13557:
13551:
13550:
13540:
13514:
13508:
13507:
13492:(2nd ed.).
13483:
13477:
13476:
13458:
13449:(5): 2089–2097.
13436:
13430:
13429:
13405:
13399:
13398:
13382:
13372:
13366:
13365:
13349:
13339:
13333:
13326:
13320:
13318:
13310:
13182:
13180:
13179:
13174:
13160:
13159:
13141:
13140:
13135:
13123:
13122:
13100:
13098:
13097:
13092:
13090:
13089:
13081:
13026:particle physics
13004:
13002:
13001:
12996:
12961:
12960:
12957:
12941:
12939:
12938:
12933:
12907:
12902:
12878:
12876:
12875:
12870:
12862:
12857:
12856:
12841:
12827:
12809:
12773:
12771:
12770:
12765:
12748:
12747:
12744:
12712:
12710:
12709:
12704:
12668:
12666:
12665:
12660:
12631:
12630:
12627:
12591:
12589:
12588:
12583:
12581:
12577:
12576:
12573:
12572:
12571:
12558:
12553:
12540:
12534:
12531:
12530:
12529:
12516:
12511:
12501:
12500:
12491:
12474:
12465:
12455:
12453:
12452:
12447:
12436:
12435:
12402:
12400:
12399:
12394:
12377:
12376:
12373:
12345:
12343:
12342:
12337:
12317:
12296:
12270:
12268:
12267:
12262:
12244:
12243:
12240:
12234:
12230:
12229:
12225:
12224:
12215:
12180:
12178:
12177:
12172:
12154:
12153:
12150:
12132:
12130:
12129:
12124:
12106:
12105:
12102:
12096:
12087:
12077:
12075:
12074:
12069:
12042:
12041:
12038:
12007:
12005:
12004:
11999:
11981:
11980:
11970:
11958:
11957:
11954:
11911:
11909:
11908:
11903:
11891:
11880:
11879:
11876:
11834:
11832:
11831:
11826:
11824:
11822:
11821:
11812:
11804:
11798:
11797:
11773:
11772:
11759:
11740:
11732:
11711:
11709:
11708:
11703:
11701:
11700:
11682:
11681:
11659:
11657:
11656:
11651:
11633:
11631:
11630:
11625:
11623:
11622:
11604:
11603:
11578:
11576:
11575:
11570:
11561:
11557:
11556:
11555:
11554:
11550:
11530:
11528:
11527:
11526:
11515:
11506:
11497:
11495:
11494:
11476:
11475:
11474:
11470:
11450:
11448:
11447:
11429:
11427:
11426:
11391:
11390:
11368:
11366:
11365:
11360:
11358:
11354:
11341:
11340:
11322:
11321:
11303:
11302:
11301:
11277:
11273:
11272:
11237:
11235:
11234:
11229:
11186:
11184:
11183:
11178:
11163:of index 1. The
11145:
11143:
11142:
11137:
11125:
11123:
11122:
11117:
11115:
11111:
11109:
11098:
11087:
11072:
11070:
11059:
11048:
11039:
11037:
11036:
11031:
10998:
10996:
10995:
10990:
10978:
10976:
10975:
10970:
10958:
10956:
10955:
10950:
10938:
10936:
10935:
10930:
10918:
10916:
10915:
10910:
10908:
10904:
10902:
10888:
10874:
10859:
10857:
10843:
10829:
10820:
10818:
10817:
10812:
10782:
10780:
10779:
10774:
10769:
10761:
10753:
10752:
10721:
10719:
10718:
10713:
10683:
10681:
10680:
10675:
10664:
10663:
10636:
10634:
10633:
10628:
10623:
10614:
10593:
10584:
10574:
10572:
10571:
10566:
10528:
10526:
10525:
10520:
10515:
10514:
10502:
10501:
10489:
10488:
10476:
10475:
10438:
10436:
10435:
10430:
10425:
10424:
10412:
10411:
10399:
10398:
10386:
10385:
10348:
10346:
10345:
10340:
10335:
10334:
10322:
10321:
10290:
10288:
10287:
10282:
10277:
10276:
10264:
10263:
10230:
10228:
10227:
10222:
10217:
10209:
10189:
10188:
10176:
10175:
10172:
10147:
10145:
10144:
10139:
10128:
10127:
10084:
10082:
10081:
10076:
10071:
10069:
10068:
10067:
10062:
10051:
10049:
10045:
10041:
10040:
10031:
10030:
10021:
10020:
10012:
10011:
10004:
10003:
9994:
9993:
9984:
9983:
9962:
9961:
9953:
9951:
9947:
9946:
9935:
9934:
9926:
9917:
9909:
9900:
9899:
9895:
9890:
9879:
9869:
9855:
9854:
9845:
9844:
9835:
9834:
9812:
9810:
9809:
9804:
9785:
9783:
9782:
9777:
9772:
9768:
9766:
9765:
9764:
9760:
9747:
9746:
9734:
9733:
9728:
9722:
9721:
9706:
9694:
9688:
9686:
9675:
9661:
9660:
9622:
9620:
9619:
9614:
9602:
9600:
9599:
9594:
9582:
9580:
9579:
9574:
9562:
9560:
9559:
9554:
9539:
9537:
9536:
9531:
9526:
9522:
9520:
9519:
9518:
9514:
9501:
9500:
9488:
9487:
9478:
9477:
9456:
9455:
9446:
9445:
9420:
9414:
9412:
9401:
9387:
9386:
9374:
9373:
9333:
9331:
9330:
9325:
9310:
9308:
9307:
9302:
9285:
9277:
9246:
9244:
9243:
9238:
9224:
9223:
9196:
9195:
9176:
9174:
9173:
9168:
9143:
9141:
9140:
9135:
9124:
9123:
9107:
9105:
9104:
9099:
9078:
9076:
9075:
9070:
9059:
9058:
9039:
9037:
9036:
9031:
9025:
9024:
8999:
8998:
8969:
8968:
8949:
8947:
8946:
8941:
8936:
8935:
8913:
8911:
8910:
8905:
8903:
8902:
8897:
8878:
8876:
8875:
8870:
8868:
8867:
8858:
8857:
8839:
8838:
8829:
8828:
8806:
8804:
8803:
8798:
8796:
8795:
8786:
8785:
8767:
8766:
8730:
8728:
8727:
8722:
8711:
8703:
8682:
8680:
8679:
8674:
8657:
8621:
8619:
8618:
8613:
8611:
8610:
8590:
8588:
8587:
8582:
8580:
8579:
8563:
8561:
8560:
8555:
8553:
8552:
8536:
8534:
8533:
8528:
8526:
8525:
8505:
8503:
8502:
8497:
8482:
8480:
8479:
8474:
8460:
8458:
8457:
8452:
8431:
8429:
8428:
8423:
8421:
8420:
8401:
8399:
8398:
8393:
8388:
8383:
8382:
8370:
8369:
8360:
8343:
8338:
8337:
8325:
8324:
8315:
8297:
8295:
8294:
8289:
8277:
8275:
8274:
8269:
8254:
8252:
8251:
8246:
8244:
8242:
8241:
8240:
8235:
8231:
8230:
8229:
8217:
8216:
8198:
8197:
8187:
8186:
8181:
8177:
8176:
8175:
8163:
8162:
8147:
8144:
8139:
8111:
8109:
8108:
8103:
8095:
8087:
8082:
8080:
8076:
8075:
8070:
8066:
8065:
8064:
8052:
8051:
8033:
8032:
8016:
8015:
8014:
8009:
8005:
8004:
8003:
7991:
7990:
7971:
7968:
7963:
7945:
7943:
7935:
7927:
7917:
7915:
7914:
7909:
7901:
7899:
7898:
7897:
7892:
7888:
7887:
7886:
7873:
7872:
7854:
7853:
7843:
7839:
7838:
7826:
7825:
7809:
7806:
7801:
7783:
7781:
7780:
7779:
7766:
7758:
7746:
7744:
7743:
7738:
7726:
7724:
7723:
7718:
7716:
7715:
7696:
7694:
7693:
7688:
7686:
7682:
7681:
7680:
7675:
7671:
7666:
7665:
7664:
7652:
7651:
7641:
7616:
7611:
7557:
7556:
7543:
7542:
7524:
7523:
7511:
7510:
7502:
7489:
7487:
7486:
7481:
7469:
7467:
7466:
7461:
7449:
7447:
7446:
7441:
7439:
7438:
7412:
7410:
7409:
7404:
7402:
7401:
7384:order statistics
7377:
7375:
7374:
7369:
7353:
7351:
7350:
7345:
7343:
7342:
7326:
7324:
7323:
7318:
7306:
7304:
7303:
7298:
7296:
7295:
7276:
7274:
7273:
7268:
7266:
7265:
7249:
7247:
7246:
7241:
7239:
7238:
7222:
7220:
7219:
7214:
7212:
7211:
7192:
7190:
7189:
7184:
7182:
7181:
7171:
7166:
7151:
7143:
7138:
7137:
7129:
7070:
7068:
7067:
7062:
7040:
7038:
7037:
7032:
7030:
7004:
6999:
6977:
6975:
6974:
6973:
6954:
6951:
6946:
6924:
6919:
6901:
6890:
6888:
6887:
6886:
6867:
6858:
6853:
6831:
6829:
6828:
6827:
6811:
6810:
6801:
6798:
6793:
6771:
6770:
6726:
6724:
6723:
6718:
6707:
6693:
6682:
6681:
6666:
6665:
6660:
6651:
6627:
6625:
6624:
6619:
6602:
6569:
6567:
6566:
6561:
6510:) cannot exist.
6499:
6497:
6496:
6491:
6466:
6461:
6445:
6423:
6421:
6420:
6415:
6390:
6385:
6372:
6327:
6325:
6324:
6319:
6300:
6291:
6289:
6288:
6283:
6258:
6253:
6218:
6213:
6193:
6178:
6169:
6167:
6166:
6161:
6133:
6128:
6108:
6104:
6102:
6101:
6096:
6075:density function
6057:
6055:
6054:
6049:
6035:
6034:
6022:
6021:
6005:
6003:
6002:
5997:
5983:
5982:
5970:
5969:
5941:
5939:
5938:
5933:
5931:
5930:
5921:
5920:
5908:
5907:
5894:
5889:
5874:
5866:
5861:
5860:
5841:
5839:
5838:
5833:
5831:
5830:
5820:
5815:
5800:
5792:
5787:
5786:
5770:
5768:
5767:
5762:
5754:
5753:
5741:
5740:
5708:
5706:
5705:
5700:
5680:
5679:
5670:
5665:
5664:
5655:
5654:
5602:
5600:
5599:
5594:
5575:
5573:
5572:
5567:
5538:
5514:
5498:
5493:
5452:
5450:
5449:
5444:
5438:
5434:
5433:
5429:
5428:
5419:
5397:
5396:
5391:
5359:
5334:
5332:
5331:
5326:
5324:
5293:
5270:
5269:
5230:
5229:
5207:
5202:
5127:
5125:
5124:
5119:
5114:
5112:
5111:
5110:
5101:
5100:
5087:
5086:
5085:
5080:
5076:
5075:
5074:
5056:
5055:
5031:
5030:
5025:
5021:
5020:
5019:
5007:
5006:
4990:
4979:
4975:
4974:
4973:
4972:
4971:
4959:
4958:
4935:
4934:
4933:
4932:
4920:
4919:
4894:
4860:
4858:
4857:
4852:
4823:
4821:
4820:
4815:
4805:
4804:
4777:
4776:
4758:
4757:
4747:
4742:
4730:
4728:
4717:
4703:
4702:
4664:
4662:
4661:
4656:
4651:
4650:
4649:
4641:
4627:
4626:
4599:
4598:
4574:
4573:
4551:
4546:
4531:
4527:
4526:
4489:
4488:
4465:
4463:
4462:
4457:
4437:
4435:
4434:
4429:
4417:
4415:
4414:
4409:
4407:
4406:
4396:
4391:
4376:
4368:
4359:
4357:
4356:
4351:
4349:
4344:
4336:
4312:
4311:
4301:
4296:
4284:
4276:
4273:
4251:
4249:
4248:
4243:
4231:
4229:
4228:
4223:
4215:
4214:
4202:
4201:
4166:
4164:
4163:
4158:
4156:
4155:
4146:
4141:
4140:
4131:
4125:
4109:
4107:
4106:
4101:
4099:
4098:
4089:
4088:
4078:
4062:
4060:
4059:
4054:
4052:
4051:
4042:
4041:
4031:
4015:
4013:
4012:
4007:
4005:
4004:
3986:
3985:
3969:
3967:
3966:
3961:
3959:
3958:
3940:
3939:
3923:
3921:
3920:
3915:
3913:
3912:
3894:
3893:
3877:
3875:
3874:
3869:
3867:
3866:
3848:
3847:
3835:
3834:
3815:
3813:
3812:
3807:
3805:
3804:
3796:
3786:
3784:
3783:
3778:
3776:
3775:
3774:
3766:
3738:
3737:
3736:
3735:
3725:
3704:
3702:
3701:
3696:
3691:
3690:
3689:
3681:
3665:
3661:
3660:
3623:
3622:
3599:
3597:
3596:
3591:
3589:
3588:
3578:
3569:
3561:
3556:
3555:
3547:
3530:
3528:
3527:
3522:
3520:
3519:
3501:
3500:
3488:
3487:
3429:
3427:
3426:
3421:
3419:
3418:
3374:
3372:
3371:
3366:
3346:
3323:
3321:
3320:
3311:
3310:
3301:
3299:
3298:
3288:
3283:
3259:) it holds that
3254:
3252:
3251:
3246:
3211:
3210:
3194:
3192:
3191:
3186:
3178:
3177:
3159:
3158:
3142:
3140:
3139:
3134:
3122:
3120:
3119:
3114:
3102:
3100:
3099:
3094:
3082:
3080:
3079:
3074:
3062:
3060:
3059:
3054:
3012:
3010:
3009:
3004:
2986:
2984:
2983:
2978:
2947:
2945:
2944:
2939:
2937:
2929:
2924:
2907:
2899:
2860:
2858:
2857:
2852:
2822:
2820:
2819:
2814:
2796:, and hence the
2795:
2793:
2792:
2787:
2776:
2775:
2757:
2756:
2734:
2732:
2731:
2726:
2721:
2717:
2716:
2712:
2711:
2702:
2667:
2666:
2645:
2644:
2602:
2600:
2599:
2594:
2592:
2584:
2579:
2575:
2570:
2569:
2568:
2552:
2540:
2532:
2518:
2517:
2475:
2473:
2472:
2467:
2461:
2459:
2448:
2433:
2431:
2430:
2425:
2410:
2408:
2407:
2402:
2397:
2393:
2391:
2390:
2389:
2377:
2376:
2367:
2366:
2347:
2346:
2337:
2325:
2323:
2319:
2318:
2317:
2312:
2308:
2303:
2302:
2301:
2285:
2264:
2244:
2243:
2209:
2207:
2206:
2201:
2195:
2193:
2189:
2188:
2163:
2120:
2118:
2117:
2112:
2094:
2092:
2091:
2086:
2078:
2077:
2058:
2056:
2055:
2050:
2048:
2044:
2042:
2031:
2023:
2017:
2016:
2013:
2009:
2001:
1996:
1992:
1990:
1976:
1970:
1969:
1966:
1962:
1954:
1922:
1920:
1919:
1914:
1903:
1902:
1877:
1875:
1874:
1869:
1867:
1866:
1844:
1842:
1841:
1836:
1834:
1832:
1821:
1784:
1782:
1781:
1776:
1760:
1758:
1757:
1752:
1733:
1731:
1730:
1725:
1709:
1707:
1706:
1701:
1699:
1698:
1679:
1677:
1676:
1671:
1666:
1662:
1660:
1659:
1658:
1646:
1645:
1636:
1635:
1613:
1607:
1599:
1594:
1592:
1591:
1587:
1586:
1585:
1580:
1576:
1571:
1570:
1569:
1553:
1525:
1511:
1510:
1467:
1465:
1464:
1459:
1454:
1439:
1437:
1436:
1431:
1404:
1402:
1401:
1396:
1391:
1367:random variables
1361:
1359:
1358:
1353:
1341:
1339:
1338:
1333:
1318:
1316:
1315:
1310:
1308:
1304:
1300:
1292:
1251:
1249:
1248:
1243:
1231:
1229:
1228:
1225:{\displaystyle }
1223:
1199:
1197:
1196:
1191:
1176:
1174:
1173:
1168:
1156:
1154:
1153:
1148:
1146:
1145:
1129:
1127:
1126:
1121:
1109:
1107:
1106:
1101:
1090:
1089:
985:upper half-plane
981:Laplace equation
928:
926:
925:
920:
909:
908:
889:
885:
883:
882:
877:
866:
865:
796:
794:
793:
788:
786:
784:
783:
782:
766:
749:
747:
746:
741:
734:
726:
706:
705:
661:
659:
658:
653:
590:
588:
587:
582:
550:
548:
547:
542:
539:
538:
514:
512:
511:
506:
503:
502:
466:
464:
463:
458:
450:
441:
410:
409:
385:
383:
382:
377:
374:
366:
361:
357:
352:
351:
350:
334:
322:
314:
297:
295:
294:
289:
286:
284:
283:
279:
278:
277:
272:
268:
263:
262:
261:
245:
216:
199:
197:
196:
191:
140:
138:
137:
132:
106:
104:
103:
98:
95:
94:
70:
56:
54:
44:
40:
21:
15948:
15947:
15943:
15942:
15941:
15939:
15938:
15937:
15898:
15897:
15896:
15891:
15863:
15839:Maximum entropy
15797:
15785:
15773:
15763:
15755:
15738:
15726:
15714:
15669:
15656:
15593:Matrix-valued:
15590:
15536:
15507:
15499:
15488:
15476:
15467:
15457:
15351:
15345:
15262:
15188:
15186:
15180:
15109:Maxwell–Jüttner
14958:Hypoexponential
14864:
14862:
14861:supported on a
14856:
14817:Noncentral beta
14777:Balding–Nichols
14759:
14758:supported on a
14750:
14740:
14643:
14637:
14633:Zipf–Mandelbrot
14563:
14554:
14548:
14538:
14457:
14454:
14449:
14444:Wayback Machine
14434:
14430:
14425:Wayback Machine
14415:
14411:
14398:
14397:
14393:
14364:
14360:
14345:
14341:
14332:
14330:
14326:
14319:
14311:
14307:
14301:Wayback Machine
14278:
14271:
14265:
14231:
14227:
14219:
14204:
14198:
14191:
14182:
14180:
14148:
14139:
14108:
14104:
14093:
14089:
14081:
14058:
14052:
14048:
14014:
14007:
13976:
13965:
13949:
13948:
13928:
13921:
13890:
13886:
13855:
13851:
13842:
13840:
13831:
13830:
13826:
13817:
13815:
13782:10.2307/3315772
13765:
13759:
13755:
13745:
13743:
13730:Kyle Siegrist.
13728:
13721:
13712:
13710:
13706:
13683:
13677:
13673:
13650:
13646:
13605:
13601:
13584:
13580:
13570:
13568:
13559:
13558:
13554:
13538:10.2307/2041858
13515:
13511:
13504:
13496:. p. 778.
13484:
13480:
13437:
13433:
13426:
13406:
13402:
13395:
13373:
13369:
13362:
13340:
13336:
13327:
13323:
13311:
13302:
13298:
13261:
13224:as part of the
13214:confidence belt
13149:
13145:
13136:
13131:
13130:
13112:
13108:
13106:
13103:
13102:
13080:
13079:
13077:
13074:
13073:
13042:
13018:
13012:
12956:
12955:
12947:
12944:
12943:
12901:
12884:
12881:
12880:
12858:
12852:
12848:
12837:
12787:
12779:
12776:
12775:
12743:
12742:
12734:
12731:
12730:
12674:
12671:
12670:
12626:
12625:
12617:
12614:
12613:
12567:
12563:
12554:
12549:
12544:
12538:
12525:
12521:
12512:
12507:
12502:
12496:
12492:
12489:
12488:
12484:
12463:
12461:
12458:
12457:
12431:
12427:
12410:
12407:
12406:
12372:
12371:
12351:
12348:
12347:
12286:
12278:
12275:
12274:
12239:
12238:
12213:
12206:
12202:
12198:
12194:
12186:
12183:
12182:
12149:
12148:
12140:
12137:
12136:
12101:
12100:
12085:
12083:
12080:
12079:
12037:
12036:
12022:
12019:
12018:
11963:
11959:
11953:
11952:
11951:
11925:
11922:
11921:
11884:
11875:
11874:
11848:
11845:
11844:
11841:
11817:
11813:
11805:
11803:
11787:
11783:
11755:
11754:
11750:
11736:
11728:
11723:
11720:
11719:
11690:
11686:
11671:
11667:
11665:
11662:
11661:
11639:
11636:
11635:
11612:
11608:
11593:
11589:
11587:
11584:
11583:
11540:
11536:
11535:
11531:
11516:
11511:
11510:
11502:
11501:
11496:
11484:
11480:
11460:
11456:
11455:
11451:
11437:
11433:
11428:
11416:
11412:
11411:
11407:
11386:
11382:
11380:
11377:
11376:
11336:
11332:
11311:
11307:
11297:
11296:
11292:
11291:
11287:
11262:
11258:
11254:
11246:
11243:
11242:
11192:
11189:
11188:
11172:
11169:
11168:
11157:
11131:
11128:
11127:
11099:
11088:
11086:
11082:
11060:
11049:
11047:
11045:
11042:
11041:
11007:
11004:
11003:
10984:
10981:
10980:
10964:
10961:
10960:
10944:
10941:
10940:
10924:
10921:
10920:
10889:
10875:
10873:
10869:
10844:
10830:
10828:
10826:
10823:
10822:
10788:
10785:
10784:
10765:
10757:
10748:
10744:
10727:
10724:
10723:
10689:
10686:
10685:
10659:
10655:
10647:
10644:
10643:
10612:
10582:
10580:
10577:
10576:
10536:
10533:
10532:
10510:
10506:
10497:
10493:
10484:
10480:
10471:
10467:
10444:
10441:
10440:
10420:
10416:
10407:
10403:
10394:
10390:
10381:
10377:
10354:
10351:
10350:
10330:
10326:
10317:
10313:
10296:
10293:
10292:
10272:
10268:
10259:
10255:
10238:
10235:
10234:
10213:
10205:
10184:
10180:
10171:
10170:
10153:
10150:
10149:
10123:
10119:
10102:
10099:
10098:
10094:
10052:
10050:
10036:
10035:
10026:
10025:
10013:
10007:
10006:
10005:
9999:
9995:
9989:
9988:
9979:
9978:
9968:
9964:
9963:
9952:
9942:
9941:
9937:
9936:
9925:
9921:
9908:
9901:
9880:
9878:
9874:
9870:
9868:
9850:
9849:
9840:
9839:
9830:
9829:
9821:
9818:
9817:
9798:
9795:
9794:
9756:
9752:
9748:
9742:
9738:
9729:
9724:
9723:
9717:
9713:
9702:
9698:
9693:
9689:
9679:
9674:
9656:
9652:
9638:
9635:
9634:
9608:
9605:
9604:
9588:
9585:
9584:
9568:
9565:
9564:
9548:
9545:
9544:
9510:
9506:
9502:
9496:
9492:
9483:
9479:
9473:
9469:
9451:
9447:
9441:
9437:
9424:
9419:
9415:
9405:
9400:
9382:
9378:
9369:
9365:
9345:
9342:
9341:
9319:
9316:
9315:
9281:
9273:
9253:
9250:
9249:
9219:
9215:
9191:
9187:
9185:
9182:
9181:
9153:
9150:
9149:
9119:
9115:
9113:
9110:
9109:
9084:
9081:
9080:
9054:
9050:
9048:
9045:
9044:
8994:
8990:
8986:
8982:
8964:
8960:
8958:
8955:
8954:
8931:
8927:
8919:
8916:
8915:
8898:
8893:
8892:
8884:
8881:
8880:
8863:
8859:
8853:
8849:
8834:
8830:
8824:
8820:
8812:
8809:
8808:
8791:
8787:
8781:
8777:
8762:
8758:
8747:
8744:
8743:
8737:
8707:
8699:
8688:
8685:
8684:
8638:
8630:
8627:
8626:
8606:
8602:
8600:
8597:
8596:
8593:Newton's method
8575:
8571:
8569:
8566:
8565:
8548:
8544:
8542:
8539:
8538:
8521:
8517:
8515:
8512:
8511:
8488:
8485:
8484:
8466:
8463:
8462:
8437:
8434:
8433:
8416:
8412:
8410:
8407:
8406:
8384:
8378:
8374:
8365:
8361:
8356:
8339:
8333:
8329:
8320:
8316:
8311:
8306:
8303:
8302:
8283:
8280:
8279:
8263:
8260:
8259:
8236:
8225:
8221:
8212:
8208:
8207:
8203:
8202:
8193:
8189:
8188:
8182:
8171:
8167:
8158:
8154:
8153:
8149:
8148:
8146:
8140:
8129:
8123:
8120:
8119:
8086:
8071:
8060:
8056:
8047:
8043:
8042:
8038:
8037:
8028:
8024:
8017:
8010:
7999:
7995:
7986:
7982:
7981:
7977:
7976:
7972:
7970:
7964:
7953:
7936:
7928:
7926:
7924:
7921:
7920:
7893:
7882:
7878:
7868:
7864:
7863:
7859:
7858:
7849:
7845:
7844:
7834:
7830:
7821:
7817:
7810:
7808:
7802:
7791:
7775:
7771:
7767:
7759:
7757:
7755:
7752:
7751:
7732:
7729:
7728:
7711:
7707:
7705:
7702:
7701:
7676:
7660:
7656:
7647:
7643:
7642:
7640:
7636:
7635:
7628:
7624:
7612:
7601:
7552:
7548:
7538:
7534:
7519:
7515:
7501:
7500:
7498:
7495:
7494:
7475:
7472:
7471:
7455:
7452:
7451:
7434:
7430:
7428:
7425:
7424:
7397:
7393:
7391:
7388:
7387:
7363:
7360:
7359:
7338:
7334:
7332:
7329:
7328:
7312:
7309:
7308:
7291:
7287:
7285:
7282:
7281:
7261:
7257:
7255:
7252:
7251:
7234:
7230:
7228:
7225:
7224:
7207:
7203:
7201:
7198:
7197:
7177:
7173:
7167:
7156:
7142:
7128:
7127:
7125:
7122:
7121:
7110:
7094:
7074:central moments
7050:
7047:
7046:
7028:
7027:
7000:
6992:
6969:
6965:
6958:
6953:
6947:
6939:
6920:
6912:
6899:
6898:
6882:
6878:
6871:
6866:
6854:
6846:
6823:
6819:
6812:
6806:
6802:
6800:
6794:
6786:
6775:
6766:
6762:
6749:
6747:
6744:
6743:
6733:
6703:
6683:
6677:
6673:
6661:
6656:
6655:
6647:
6636:
6633:
6632:
6583:
6575:
6572:
6571:
6534:
6531:
6530:
6527:
6525:Smaller moments
6515:expected values
6462:
6451:
6435:
6429:
6426:
6425:
6386:
6378:
6362:
6356:
6353:
6352:
6313:
6310:
6309:
6254:
6249:
6214:
6206:
6200:
6197:
6196:
6129:
6121:
6115:
6112:
6111:
6081:
6078:
6077:
6067:
6030:
6026:
6017:
6013:
6011:
6008:
6007:
5978:
5974:
5965:
5961:
5959:
5956:
5955:
5926:
5922:
5916:
5912:
5903:
5899:
5890:
5879:
5865:
5856:
5852:
5850:
5847:
5846:
5826:
5822:
5816:
5805:
5791:
5782:
5778:
5776:
5773:
5772:
5749:
5745:
5736:
5732:
5730:
5727:
5726:
5723:
5715:
5675:
5671:
5666:
5660:
5656:
5650:
5646:
5611:
5608:
5607:
5588:
5585:
5584:
5534:
5507:
5494:
5489:
5468:
5465:
5464:
5417:
5410:
5406:
5402:
5398:
5392:
5387:
5386:
5352:
5350:
5347:
5346:
5322:
5321:
5291:
5290:
5265:
5261:
5225:
5221:
5203:
5195:
5181:
5165:
5163:
5160:
5159:
5153:
5137:total variation
5106:
5102:
5096:
5092:
5088:
5081:
5064:
5060:
5045:
5041:
5040:
5036:
5035:
5026:
5015:
5011:
5002:
4998:
4997:
4993:
4992:
4991:
4989:
4967:
4963:
4948:
4944:
4943:
4939:
4928:
4924:
4909:
4905:
4904:
4900:
4899:
4895:
4887:
4885:
4882:
4881:
4871:
4840:
4837:
4836:
4791:
4787:
4772:
4768:
4753:
4749:
4743:
4735:
4721:
4716:
4698:
4694:
4680:
4677:
4676:
4645:
4637:
4622:
4618:
4614:
4610:
4588:
4584:
4569:
4565:
4547:
4539:
4516:
4512:
4508:
4484:
4480:
4478:
4475:
4474:
4451:
4448:
4447:
4444:
4423:
4420:
4419:
4402:
4398:
4392:
4381:
4367:
4365:
4362:
4361:
4337:
4335:
4307:
4303:
4297:
4289:
4275:
4263:
4257:
4254:
4253:
4237:
4234:
4233:
4210:
4206:
4197:
4193:
4191:
4188:
4187:
4184:
4151:
4147:
4142:
4136:
4132:
4127:
4121:
4115:
4112:
4111:
4094:
4090:
4084:
4080:
4074:
4068:
4065:
4064:
4047:
4043:
4037:
4033:
4027:
4021:
4018:
4017:
4000:
3996:
3981:
3977:
3975:
3972:
3971:
3954:
3950:
3935:
3931:
3929:
3926:
3925:
3908:
3904:
3889:
3885:
3883:
3880:
3879:
3862:
3858:
3843:
3839:
3830:
3826:
3824:
3821:
3820:
3795:
3794:
3792:
3789:
3788:
3770:
3762:
3755:
3751:
3731:
3727:
3721:
3716:
3712:
3710:
3707:
3706:
3685:
3677:
3673:
3669:
3650:
3646:
3642:
3618:
3614:
3612:
3609:
3608:
3584:
3580:
3574:
3560:
3546:
3545:
3543:
3540:
3539:
3515:
3511:
3496:
3492:
3483:
3479:
3477:
3474:
3473:
3470:
3414:
3410:
3408:
3405:
3404:
3381:
3327:
3316:
3312:
3306:
3302:
3300:
3294:
3290:
3284:
3273:
3267:
3264:
3263:
3206:
3202:
3200:
3197:
3196:
3173:
3169:
3154:
3150:
3148:
3145:
3144:
3128:
3125:
3124:
3108:
3105:
3104:
3103:independent of
3088:
3085:
3084:
3068:
3065:
3064:
3063:and any random
3021:
3018:
3017:
2992:
2989:
2988:
2972:
2969:
2968:
2954:
2928:
2914:
2898:
2869:
2866:
2865:
2834:
2831:
2830:
2805:
2802:
2801:
2771:
2767:
2752:
2748:
2743:
2740:
2739:
2700:
2693:
2689:
2685:
2681:
2662:
2658:
2640:
2636:
2622:
2619:
2618:
2583:
2564:
2560:
2553:
2551:
2547:
2531:
2513:
2509:
2495:
2492:
2491:
2481:
2452:
2447:
2439:
2436:
2435:
2419:
2416:
2415:
2385:
2381:
2372:
2368:
2362:
2358:
2348:
2342:
2338:
2336:
2332:
2313:
2297:
2293:
2286:
2284:
2280:
2279:
2272:
2268:
2263:
2239:
2235:
2221:
2218:
2217:
2184:
2180:
2167:
2162:
2133:
2130:
2129:
2100:
2097:
2096:
2073:
2069:
2067:
2064:
2063:
2032:
2024:
2022:
2018:
2012:
2011:
2000:
1980:
1975:
1971:
1965:
1964:
1953:
1930:
1927:
1926:
1898:
1894:
1886:
1883:
1882:
1862:
1858:
1850:
1847:
1846:
1825:
1820:
1818:
1815:
1814:
1811:
1770:
1767:
1766:
1743:
1740:
1739:
1736:scale parameter
1719:
1716:
1715:
1694:
1690:
1688:
1685:
1684:
1654:
1650:
1641:
1637:
1631:
1627:
1617:
1612:
1608:
1598:
1581:
1565:
1561:
1554:
1552:
1548:
1547:
1540:
1536:
1529:
1524:
1506:
1502:
1488:
1485:
1484:
1474:
1450:
1445:
1442:
1441:
1413:
1410:
1409:
1387:
1382:
1379:
1378:
1347:
1344:
1343:
1327:
1324:
1323:
1291:
1278:
1274:
1260:
1257:
1256:
1237:
1234:
1233:
1205:
1202:
1201:
1185:
1182:
1181:
1162:
1159:
1158:
1141:
1137:
1135:
1132:
1131:
1115:
1112:
1111:
1085:
1081:
1076:
1073:
1072:
1065:
1057:
1028:witch of Agnesi
1008:
975:, which is the
904:
900:
895:
892:
891:
887:
861:
857:
843:
840:
839:
824:Hendrik Lorentz
808:Augustin Cauchy
778:
774:
770:
765:
763:
760:
759:
730:
722:
701:
697:
685:
682:
681:
628:
625:
624:
608:Excess kurtosis
576:
573:
572:
534:
530:
528:
525:
524:
498:
494:
492:
489:
488:
439:
405:
401:
399:
396:
395:
365:
346:
342:
335:
333:
329:
313:
311:
308:
307:
273:
257:
253:
246:
244:
240:
239:
232:
228:
220:
215:
213:
210:
209:
158:
155:
154:
120:
117:
116:
115:
90:
86:
84:
81:
80:
65:
55:
49:
39:
28:
23:
22:
15:
12:
11:
5:
15946:
15936:
15935:
15930:
15925:
15920:
15915:
15910:
15893:
15892:
15890:
15889:
15879:
15868:
15865:
15864:
15862:
15861:
15856:
15851:
15846:
15841:
15836:
15834:Location–scale
15831:
15826:
15821:
15816:
15811:
15805:
15803:
15799:
15798:
15796:
15795:
15790:
15783:
15778:
15770:
15768:
15757:
15756:
15754:
15753:
15748:
15743:
15736:
15731:
15724:
15719:
15712:
15707:
15702:
15697:
15695:Wrapped Cauchy
15692:
15690:Wrapped normal
15687:
15682:
15677:
15666:
15664:
15658:
15657:
15655:
15654:
15653:
15652:
15647:
15645:Normal-inverse
15642:
15637:
15627:
15626:
15625:
15615:
15607:
15602:
15597:
15588:
15587:
15586:
15576:
15568:
15563:
15558:
15553:
15552:
15551:
15541:
15534:
15533:
15532:
15527:
15517:
15512:
15504:
15502:
15494:
15493:
15490:
15489:
15487:
15486:
15480:
15478:
15469:
15463:
15462:
15459:
15458:
15456:
15455:
15450:
15445:
15437:
15429:
15421:
15412:
15403:
15394:
15385:
15376:
15371:
15366:
15361:
15355:
15353:
15347:
15346:
15344:
15343:
15338:
15336:Variance-gamma
15333:
15328:
15320:
15315:
15310:
15305:
15300:
15295:
15287:
15282:
15281:
15280:
15270:
15265:
15260:
15254:
15249:
15244:
15239:
15234:
15229:
15224:
15216:
15211:
15203:
15198:
15192:
15190:
15182:
15181:
15179:
15178:
15176:Wilks's lambda
15173:
15172:
15171:
15161:
15156:
15151:
15146:
15141:
15136:
15131:
15126:
15121:
15116:
15114:Mittag-Leffler
15111:
15106:
15101:
15096:
15091:
15086:
15081:
15076:
15071:
15066:
15061:
15056:
15055:
15054:
15044:
15035:
15030:
15025:
15024:
15023:
15013:
15011:gamma/Gompertz
15008:
15007:
15006:
15001:
14991:
14986:
14981:
14980:
14979:
14967:
14966:
14965:
14960:
14955:
14945:
14944:
14943:
14933:
14928:
14923:
14922:
14921:
14920:
14919:
14909:
14899:
14894:
14889:
14884:
14879:
14874:
14868:
14866:
14863:semi-infinite
14858:
14857:
14855:
14854:
14849:
14844:
14839:
14834:
14829:
14824:
14819:
14814:
14809:
14804:
14799:
14794:
14789:
14784:
14779:
14774:
14769:
14763:
14761:
14752:
14746:
14745:
14742:
14741:
14739:
14738:
14733:
14728:
14723:
14718:
14713:
14708:
14703:
14698:
14693:
14688:
14683:
14678:
14673:
14668:
14663:
14658:
14653:
14647:
14645:
14642:with infinite
14639:
14638:
14636:
14635:
14630:
14625:
14620:
14615:
14610:
14605:
14604:
14603:
14596:Hypergeometric
14593:
14588:
14583:
14578:
14573:
14567:
14565:
14556:
14550:
14549:
14537:
14536:
14529:
14522:
14514:
14508:
14507:
14502:
14497:
14478:
14473:
14453:
14452:External links
14450:
14448:
14447:
14428:
14409:
14391:
14358:
14355:. p. 603.
14353:Addison-Wesley
14339:
14322:. p. 11.
14305:
14269:
14263:
14225:
14222:on 2009-09-14.
14189:
14137:
14102:
14087:
14084:on 2011-08-16.
14046:
14005:
13963:
13919:
13884:
13849:
13824:
13776:(3): 497–513,
13753:
13719:
13671:
13644:
13599:
13578:
13552:
13531:(1): 130–135.
13509:
13502:
13478:
13431:
13424:
13400:
13393:
13367:
13360:
13334:
13321:
13299:
13297:
13294:
13293:
13292:
13287:
13285:Stable process
13282:
13280:Cauchy process
13277:
13271:
13260:
13257:
13256:
13255:
13236:
13229:
13205:
13185:
13184:
13172:
13169:
13166:
13163:
13158:
13155:
13152:
13148:
13144:
13139:
13134:
13129:
13126:
13121:
13118:
13115:
13111:
13087:
13084:
13066:
13051:spectral lines
13041:
13038:
13014:Main article:
13011:
13008:
13007:
13006:
12994:
12991:
12988:
12985:
12982:
12979:
12976:
12973:
12970:
12967:
12964:
12954:
12951:
12931:
12928:
12925:
12922:
12919:
12916:
12913:
12910:
12905:
12900:
12897:
12894:
12891:
12888:
12868:
12865:
12861:
12855:
12851:
12847:
12844:
12840:
12836:
12833:
12830:
12826:
12823:
12820:
12817:
12814:
12808:
12805:
12802:
12799:
12796:
12793:
12790:
12786:
12783:
12763:
12760:
12757:
12754:
12751:
12741:
12738:
12727:
12720:
12714:
12702:
12699:
12696:
12693:
12690:
12687:
12684:
12681:
12678:
12658:
12655:
12652:
12649:
12646:
12643:
12640:
12637:
12634:
12624:
12621:
12606:
12599:
12592:
12580:
12570:
12566:
12562:
12557:
12552:
12548:
12543:
12537:
12528:
12524:
12520:
12515:
12510:
12506:
12499:
12495:
12487:
12483:
12480:
12477:
12471:
12468:
12445:
12442:
12439:
12434:
12430:
12426:
12423:
12420:
12417:
12414:
12403:
12392:
12389:
12386:
12383:
12380:
12370:
12367:
12364:
12361:
12358:
12355:
12335:
12332:
12329:
12326:
12323:
12320:
12316:
12313:
12310:
12307:
12304:
12301:
12295:
12292:
12289:
12285:
12282:
12271:
12259:
12256:
12253:
12250:
12247:
12237:
12233:
12228:
12221:
12218:
12212:
12209:
12205:
12201:
12197:
12193:
12190:
12169:
12166:
12163:
12160:
12157:
12147:
12144:
12133:
12121:
12118:
12115:
12112:
12109:
12099:
12093:
12090:
12067:
12064:
12061:
12057:
12054:
12051:
12048:
12045:
12035:
12032:
12029:
12026:
12015:
11996:
11993:
11990:
11987:
11984:
11979:
11976:
11973:
11969:
11966:
11962:
11950:
11947:
11944:
11941:
11938:
11935:
11932:
11929:
11919:
11900:
11897:
11894:
11890:
11887:
11883:
11873:
11870:
11867:
11864:
11861:
11858:
11855:
11852:
11840:
11837:
11836:
11835:
11820:
11816:
11811:
11808:
11801:
11796:
11793:
11790:
11786:
11782:
11779:
11776:
11771:
11768:
11765:
11762:
11758:
11753:
11749:
11746:
11743:
11739:
11735:
11731:
11727:
11699:
11696:
11693:
11689:
11685:
11680:
11677:
11674:
11670:
11649:
11646:
11643:
11621:
11618:
11615:
11611:
11607:
11602:
11599:
11596:
11592:
11580:
11579:
11568:
11565:
11560:
11553:
11549:
11546:
11543:
11539:
11534:
11525:
11522:
11519:
11514:
11509:
11505:
11500:
11493:
11490:
11487:
11483:
11479:
11473:
11469:
11466:
11463:
11459:
11454:
11446:
11443:
11440:
11436:
11432:
11425:
11422:
11419:
11415:
11410:
11406:
11403:
11400:
11397:
11394:
11389:
11385:
11370:
11369:
11357:
11353:
11350:
11347:
11344:
11339:
11335:
11331:
11328:
11325:
11320:
11317:
11314:
11310:
11306:
11300:
11295:
11290:
11286:
11283:
11280:
11276:
11271:
11268:
11265:
11261:
11257:
11253:
11250:
11226:
11223:
11220:
11217:
11214:
11211:
11208:
11205:
11202:
11199:
11196:
11187:is given, for
11176:
11156:
11153:
11152:
11151:
11135:
11114:
11108:
11105:
11102:
11097:
11094:
11091:
11085:
11081:
11078:
11075:
11069:
11066:
11063:
11058:
11055:
11052:
11029:
11026:
11023:
11020:
11017:
11014:
11011:
11000:
10988:
10968:
10948:
10928:
10907:
10901:
10898:
10895:
10892:
10887:
10884:
10881:
10878:
10872:
10868:
10865:
10862:
10856:
10853:
10850:
10847:
10842:
10839:
10836:
10833:
10810:
10807:
10804:
10801:
10798:
10795:
10792:
10772:
10768:
10764:
10760:
10756:
10751:
10747:
10743:
10740:
10737:
10734:
10731:
10711:
10708:
10705:
10702:
10699:
10696:
10693:
10673:
10670:
10667:
10662:
10658:
10654:
10651:
10637:
10626:
10620:
10617:
10611:
10608:
10605:
10602:
10599:
10596:
10590:
10587:
10564:
10561:
10558:
10555:
10552:
10549:
10546:
10543:
10540:
10529:
10518:
10513:
10509:
10505:
10500:
10496:
10492:
10487:
10483:
10479:
10474:
10470:
10466:
10463:
10460:
10457:
10454:
10451:
10448:
10428:
10423:
10419:
10415:
10410:
10406:
10402:
10397:
10393:
10389:
10384:
10380:
10376:
10373:
10370:
10367:
10364:
10361:
10358:
10338:
10333:
10329:
10325:
10320:
10316:
10312:
10309:
10306:
10303:
10300:
10280:
10275:
10271:
10267:
10262:
10258:
10254:
10251:
10248:
10245:
10242:
10231:
10220:
10216:
10212:
10208:
10204:
10201:
10198:
10195:
10192:
10187:
10183:
10179:
10169:
10166:
10163:
10160:
10157:
10137:
10134:
10131:
10126:
10122:
10118:
10115:
10112:
10109:
10106:
10093:
10090:
10086:
10085:
10074:
10065:
10061:
10058:
10055:
10048:
10044:
10039:
10034:
10029:
10024:
10019:
10016:
10010:
10002:
9998:
9992:
9987:
9982:
9977:
9974:
9971:
9967:
9959:
9956:
9950:
9945:
9940:
9932:
9929:
9924:
9920:
9915:
9912:
9907:
9904:
9898:
9893:
9889:
9886:
9883:
9877:
9873:
9867:
9864:
9861:
9858:
9853:
9848:
9843:
9838:
9833:
9828:
9825:
9802:
9787:
9786:
9775:
9771:
9763:
9759:
9755:
9751:
9745:
9741:
9737:
9732:
9727:
9720:
9716:
9712:
9709:
9705:
9701:
9697:
9692:
9685:
9682:
9678:
9673:
9670:
9667:
9664:
9659:
9655:
9651:
9648:
9645:
9642:
9612:
9592:
9572:
9552:
9541:
9540:
9529:
9525:
9517:
9513:
9509:
9505:
9499:
9495:
9491:
9486:
9482:
9476:
9472:
9468:
9465:
9462:
9459:
9454:
9450:
9444:
9440:
9436:
9433:
9430:
9427:
9423:
9418:
9411:
9408:
9404:
9399:
9396:
9393:
9390:
9385:
9381:
9377:
9372:
9368:
9364:
9361:
9358:
9355:
9352:
9349:
9323:
9312:
9311:
9300:
9297:
9294:
9291:
9288:
9284:
9280:
9276:
9272:
9269:
9266:
9263:
9260:
9257:
9247:
9236:
9233:
9230:
9227:
9222:
9218:
9214:
9211:
9208:
9205:
9202:
9199:
9194:
9190:
9166:
9163:
9160:
9157:
9133:
9130:
9127:
9122:
9118:
9097:
9094:
9091:
9088:
9068:
9065:
9062:
9057:
9053:
9041:
9040:
9028:
9023:
9020:
9017:
9014:
9011:
9008:
9005:
9002:
8997:
8993:
8989:
8985:
8981:
8978:
8975:
8972:
8967:
8963:
8939:
8934:
8930:
8926:
8923:
8901:
8896:
8891:
8888:
8866:
8862:
8856:
8852:
8848:
8845:
8842:
8837:
8833:
8827:
8823:
8819:
8816:
8794:
8790:
8784:
8780:
8776:
8773:
8770:
8765:
8761:
8757:
8754:
8751:
8736:
8733:
8720:
8717:
8714:
8710:
8706:
8702:
8698:
8695:
8692:
8672:
8669:
8666:
8663:
8660:
8656:
8653:
8650:
8647:
8644:
8641:
8637:
8634:
8609:
8605:
8578:
8574:
8551:
8547:
8524:
8520:
8495:
8492:
8472:
8450:
8447:
8444:
8441:
8419:
8415:
8403:
8402:
8391:
8387:
8381:
8377:
8373:
8368:
8364:
8359:
8355:
8352:
8349:
8346:
8342:
8336:
8332:
8328:
8323:
8319:
8314:
8310:
8287:
8267:
8256:
8255:
8239:
8234:
8228:
8224:
8220:
8215:
8211:
8206:
8201:
8196:
8192:
8185:
8180:
8174:
8170:
8166:
8161:
8157:
8152:
8143:
8138:
8135:
8132:
8128:
8113:
8112:
8101:
8098:
8093:
8090:
8085:
8079:
8074:
8069:
8063:
8059:
8055:
8050:
8046:
8041:
8036:
8031:
8027:
8023:
8020:
8013:
8008:
8002:
7998:
7994:
7989:
7985:
7980:
7975:
7967:
7962:
7959:
7956:
7952:
7948:
7942:
7939:
7934:
7931:
7918:
7907:
7904:
7896:
7891:
7885:
7881:
7876:
7871:
7867:
7862:
7857:
7852:
7848:
7842:
7837:
7833:
7829:
7824:
7820:
7816:
7813:
7805:
7800:
7797:
7794:
7790:
7786:
7778:
7774:
7770:
7765:
7762:
7736:
7714:
7710:
7698:
7697:
7685:
7679:
7674:
7669:
7663:
7659:
7655:
7650:
7646:
7639:
7634:
7631:
7627:
7623:
7620:
7615:
7610:
7607:
7604:
7600:
7596:
7593:
7590:
7587:
7584:
7581:
7578:
7575:
7572:
7569:
7566:
7563:
7560:
7555:
7551:
7546:
7541:
7537:
7533:
7530:
7527:
7522:
7518:
7514:
7508:
7505:
7479:
7459:
7437:
7433:
7400:
7396:
7380:truncated mean
7367:
7341:
7337:
7316:
7294:
7290:
7264:
7260:
7237:
7233:
7210:
7206:
7194:
7193:
7180:
7176:
7170:
7165:
7162:
7159:
7155:
7149:
7146:
7141:
7135:
7132:
7109:
7106:
7093:
7090:
7060:
7057:
7054:
7042:
7041:
7026:
7023:
7020:
7017:
7014:
7011:
7008:
7003:
6998:
6995:
6991:
6987:
6984:
6981:
6972:
6968:
6964:
6961:
6957:
6950:
6945:
6942:
6938:
6934:
6931:
6928:
6923:
6918:
6915:
6911:
6907:
6904:
6902:
6900:
6897:
6894:
6885:
6881:
6877:
6874:
6870:
6865:
6862:
6857:
6852:
6849:
6845:
6841:
6838:
6835:
6826:
6822:
6818:
6815:
6809:
6805:
6797:
6792:
6789:
6785:
6781:
6778:
6776:
6774:
6769:
6765:
6761:
6758:
6755:
6752:
6751:
6732:
6731:Higher moments
6729:
6728:
6727:
6716:
6713:
6710:
6706:
6702:
6699:
6696:
6692:
6689:
6686:
6680:
6676:
6672:
6669:
6664:
6659:
6654:
6650:
6646:
6643:
6640:
6617:
6614:
6611:
6608:
6605:
6601:
6598:
6595:
6592:
6589:
6586:
6582:
6579:
6559:
6556:
6553:
6550:
6547:
6544:
6541:
6538:
6526:
6523:
6517:, such as the
6489:
6486:
6482:
6479:
6476:
6473:
6470:
6465:
6460:
6457:
6454:
6450:
6444:
6441:
6438:
6434:
6413:
6410:
6406:
6403:
6400:
6397:
6394:
6389:
6384:
6381:
6377:
6371:
6368:
6365:
6361:
6347:Note that the
6317:
6304:
6303:
6294:
6292:
6281:
6278:
6274:
6271:
6268:
6265:
6262:
6257:
6252:
6248:
6244:
6241:
6238:
6234:
6231:
6228:
6225:
6222:
6217:
6212:
6209:
6205:
6182:
6181:
6172:
6170:
6159:
6156:
6153:
6149:
6146:
6143:
6140:
6137:
6132:
6127:
6124:
6120:
6094:
6091:
6088:
6085:
6066:
6063:
6047:
6044:
6041:
6038:
6033:
6029:
6025:
6020:
6016:
5995:
5992:
5989:
5986:
5981:
5977:
5973:
5968:
5964:
5929:
5925:
5919:
5915:
5911:
5906:
5902:
5898:
5893:
5888:
5885:
5882:
5878:
5872:
5869:
5864:
5859:
5855:
5829:
5825:
5819:
5814:
5811:
5808:
5804:
5798:
5795:
5790:
5785:
5781:
5760:
5757:
5752:
5748:
5744:
5739:
5735:
5722:
5721:Sample moments
5719:
5714:
5711:
5710:
5709:
5698:
5695:
5692:
5689:
5686:
5683:
5678:
5674:
5669:
5663:
5659:
5653:
5649:
5645:
5642:
5639:
5636:
5633:
5630:
5627:
5624:
5621:
5618:
5615:
5592:
5577:
5576:
5565:
5562:
5559:
5556:
5553:
5550:
5547:
5544:
5541:
5537:
5532:
5529:
5526:
5523:
5520:
5517:
5513:
5510:
5506:
5502:
5497:
5492:
5488:
5484:
5481:
5478:
5475:
5472:
5454:
5453:
5441:
5437:
5432:
5425:
5422:
5416:
5413:
5409:
5405:
5401:
5395:
5390:
5384:
5380:
5377:
5374:
5371:
5368:
5365:
5362:
5358:
5355:
5336:
5335:
5320:
5317:
5314:
5311:
5308:
5305:
5302:
5299:
5296:
5294:
5292:
5289:
5286:
5282:
5279:
5276:
5273:
5268:
5264:
5260:
5257:
5254:
5251:
5248:
5245:
5242:
5239:
5236:
5233:
5228:
5224:
5220:
5217:
5214:
5211:
5206:
5201:
5198:
5194:
5190:
5187:
5184:
5182:
5180:
5177:
5174:
5171:
5168:
5167:
5152:
5149:
5129:
5128:
5117:
5109:
5105:
5099:
5095:
5091:
5084:
5079:
5073:
5070:
5067:
5063:
5059:
5054:
5051:
5048:
5044:
5039:
5034:
5029:
5024:
5018:
5014:
5010:
5005:
5001:
4996:
4988:
4985:
4982:
4978:
4970:
4966:
4962:
4957:
4954:
4951:
4947:
4942:
4938:
4931:
4927:
4923:
4918:
4915:
4912:
4908:
4903:
4898:
4893:
4890:
4870:
4867:
4863:differentiable
4850:
4847:
4844:
4825:
4824:
4812:
4809:
4803:
4800:
4797:
4794:
4790:
4786:
4783:
4780:
4775:
4771:
4767:
4764:
4761:
4756:
4752:
4746:
4741:
4738:
4734:
4727:
4724:
4720:
4715:
4712:
4709:
4706:
4701:
4697:
4693:
4690:
4687:
4684:
4666:
4665:
4654:
4648:
4644:
4640:
4636:
4633:
4630:
4625:
4621:
4617:
4613:
4609:
4606:
4603:
4597:
4594:
4591:
4587:
4583:
4580:
4577:
4572:
4568:
4564:
4561:
4558:
4555:
4550:
4545:
4542:
4538:
4534:
4530:
4525:
4522:
4519:
4515:
4511:
4507:
4504:
4501:
4498:
4495:
4492:
4487:
4483:
4455:
4443:
4440:
4427:
4405:
4401:
4395:
4390:
4387:
4384:
4380:
4374:
4371:
4347:
4343:
4340:
4334:
4331:
4328:
4324:
4321:
4318:
4315:
4310:
4306:
4300:
4295:
4292:
4288:
4282:
4279:
4272:
4269:
4266:
4262:
4241:
4221:
4218:
4213:
4209:
4205:
4200:
4196:
4183:
4180:
4154:
4150:
4145:
4139:
4135:
4130:
4124:
4120:
4097:
4093:
4087:
4083:
4077:
4073:
4050:
4046:
4040:
4036:
4030:
4026:
4003:
3999:
3995:
3992:
3989:
3984:
3980:
3957:
3953:
3949:
3946:
3943:
3938:
3934:
3911:
3907:
3903:
3900:
3897:
3892:
3888:
3865:
3861:
3857:
3854:
3851:
3846:
3842:
3838:
3833:
3829:
3802:
3799:
3773:
3769:
3765:
3761:
3758:
3754:
3750:
3747:
3744:
3741:
3734:
3730:
3724:
3720:
3715:
3694:
3688:
3684:
3680:
3676:
3672:
3668:
3664:
3659:
3656:
3653:
3649:
3645:
3641:
3638:
3635:
3632:
3629:
3626:
3621:
3617:
3587:
3583:
3577:
3573:
3567:
3564:
3559:
3553:
3550:
3518:
3514:
3510:
3507:
3504:
3499:
3495:
3491:
3486:
3482:
3469:
3466:
3441:distribution.
3417:
3413:
3380:
3377:
3376:
3375:
3364:
3361:
3358:
3355:
3352:
3349:
3345:
3342:
3339:
3336:
3333:
3330:
3326:
3319:
3315:
3309:
3305:
3297:
3293:
3287:
3282:
3279:
3276:
3272:
3244:
3241:
3238:
3235:
3232:
3229:
3226:
3223:
3220:
3217:
3214:
3209:
3205:
3184:
3181:
3176:
3172:
3168:
3165:
3162:
3157:
3153:
3132:
3112:
3092:
3072:
3052:
3049:
3046:
3043:
3040:
3037:
3034:
3031:
3028:
3025:
3002:
2999:
2996:
2976:
2953:
2950:
2949:
2948:
2935:
2932:
2927:
2923:
2920:
2917:
2913:
2910:
2905:
2902:
2897:
2894:
2891:
2888:
2885:
2882:
2879:
2876:
2873:
2850:
2847:
2844:
2841:
2838:
2812:
2809:
2785:
2782:
2779:
2774:
2770:
2766:
2763:
2760:
2755:
2751:
2747:
2736:
2735:
2724:
2720:
2715:
2708:
2705:
2699:
2696:
2692:
2688:
2684:
2680:
2677:
2673:
2670:
2665:
2661:
2657:
2654:
2651:
2648:
2643:
2639:
2635:
2632:
2629:
2626:
2604:
2603:
2590:
2587:
2582:
2578:
2573:
2567:
2563:
2559:
2556:
2550:
2546:
2543:
2538:
2535:
2530:
2527:
2524:
2521:
2516:
2512:
2508:
2505:
2502:
2499:
2480:
2477:
2464:
2458:
2455:
2451:
2446:
2443:
2423:
2412:
2411:
2400:
2396:
2388:
2384:
2380:
2375:
2371:
2365:
2361:
2357:
2354:
2351:
2345:
2341:
2335:
2331:
2328:
2322:
2316:
2311:
2306:
2300:
2296:
2292:
2289:
2283:
2278:
2275:
2271:
2267:
2262:
2259:
2256:
2253:
2250:
2247:
2242:
2238:
2234:
2231:
2228:
2225:
2211:
2210:
2198:
2192:
2187:
2183:
2179:
2176:
2173:
2170:
2166:
2161:
2158:
2155:
2152:
2149:
2146:
2143:
2140:
2137:
2121:is called the
2110:
2107:
2104:
2084:
2081:
2076:
2072:
2060:
2059:
2047:
2041:
2038:
2035:
2030:
2027:
2021:
2007:
2004:
1999:
1995:
1989:
1986:
1983:
1979:
1974:
1960:
1957:
1952:
1949:
1946:
1943:
1940:
1937:
1934:
1912:
1909:
1906:
1901:
1897:
1893:
1890:
1865:
1861:
1857:
1854:
1831:
1828:
1824:
1810:
1807:
1791:probable error
1774:
1750:
1747:
1723:
1697:
1693:
1681:
1680:
1669:
1665:
1657:
1653:
1649:
1644:
1640:
1634:
1630:
1626:
1623:
1620:
1616:
1611:
1605:
1602:
1597:
1590:
1584:
1579:
1574:
1568:
1564:
1560:
1557:
1551:
1546:
1543:
1539:
1535:
1532:
1528:
1523:
1520:
1517:
1514:
1509:
1505:
1501:
1498:
1495:
1492:
1473:
1470:
1457:
1453:
1449:
1429:
1426:
1423:
1420:
1417:
1394:
1390:
1386:
1371:expected value
1351:
1331:
1320:
1319:
1307:
1303:
1298:
1295:
1290:
1287:
1284:
1281:
1277:
1273:
1270:
1267:
1264:
1241:
1221:
1218:
1215:
1212:
1209:
1189:
1166:
1144:
1140:
1119:
1099:
1096:
1093:
1088:
1084:
1080:
1064:
1061:
1056:
1053:
1043:'s use of the
1007:
1004:
973:Poisson kernel
954:§ Moments
946:expected value
918:
915:
912:
907:
903:
899:
875:
872:
869:
864:
860:
856:
853:
850:
847:
806:, named after
798:
797:
781:
777:
773:
769:
757:
751:
750:
737:
733:
729:
725:
720:
717:
714:
710:
704:
700:
696:
693:
690:
679:
673:
672:
671:does not exist
669:
663:
662:
650:
647:
644:
641:
638:
635:
632:
622:
616:
615:
610:
604:
603:
598:
592:
591:
580:
570:
564:
563:
558:
552:
551:
537:
533:
522:
516:
515:
501:
497:
486:
480:
479:
474:
468:
467:
456:
453:
447:
444:
438:
435:
432:
429:
426:
423:
420:
416:
413:
408:
404:
393:
387:
386:
372:
369:
364:
360:
355:
349:
345:
341:
338:
332:
328:
325:
320:
317:
305:
299:
298:
282:
276:
271:
266:
260:
256:
252:
249:
243:
238:
235:
231:
226:
223:
219:
207:
201:
200:
187:
184:
181:
178:
175:
172:
169:
166:
163:
152:
146:
145:
130:
127:
124:
93:
89:
78:
72:
71:
63:
60:
59:
47:
26:
9:
6:
4:
3:
2:
15945:
15934:
15931:
15929:
15926:
15924:
15921:
15919:
15916:
15914:
15911:
15909:
15906:
15905:
15903:
15888:
15880:
15878:
15870:
15869:
15866:
15860:
15857:
15855:
15852:
15850:
15847:
15845:
15842:
15840:
15837:
15835:
15832:
15830:
15827:
15825:
15822:
15820:
15817:
15815:
15812:
15810:
15807:
15806:
15804:
15800:
15794:
15791:
15788:
15784:
15782:
15779:
15776:
15772:
15771:
15769:
15767:
15762:
15758:
15752:
15749:
15747:
15744:
15741:
15737:
15735:
15732:
15729:
15725:
15723:
15720:
15717:
15713:
15711:
15708:
15706:
15703:
15701:
15698:
15696:
15693:
15691:
15688:
15686:
15683:
15681:
15678:
15675:
15674:
15668:
15667:
15665:
15663:
15659:
15651:
15648:
15646:
15643:
15641:
15638:
15636:
15633:
15632:
15631:
15628:
15624:
15621:
15620:
15619:
15616:
15614:
15613:
15608:
15606:
15605:Matrix normal
15603:
15601:
15598:
15595:
15594:
15589:
15585:
15582:
15581:
15580:
15577:
15575:
15574:
15571:Multivariate
15569:
15567:
15564:
15562:
15559:
15557:
15554:
15550:
15547:
15546:
15545:
15542:
15539:
15535:
15531:
15528:
15526:
15523:
15522:
15521:
15518:
15516:
15513:
15510:
15506:
15505:
15503:
15501:
15498:Multivariate
15495:
15485:
15482:
15481:
15479:
15473:
15470:
15464:
15454:
15451:
15449:
15446:
15444:
15442:
15438:
15436:
15434:
15430:
15428:
15426:
15422:
15420:
15418:
15413:
15411:
15409:
15404:
15402:
15400:
15395:
15393:
15391:
15386:
15384:
15382:
15377:
15375:
15372:
15370:
15367:
15365:
15362:
15360:
15357:
15356:
15354:
15350:with support
15348:
15342:
15339:
15337:
15334:
15332:
15329:
15327:
15326:
15321:
15319:
15316:
15314:
15311:
15309:
15306:
15304:
15301:
15299:
15296:
15294:
15293:
15288:
15286:
15283:
15279:
15276:
15275:
15274:
15271:
15269:
15266:
15264:
15263:
15255:
15253:
15250:
15248:
15245:
15243:
15240:
15238:
15235:
15233:
15230:
15228:
15225:
15223:
15222:
15217:
15215:
15212:
15210:
15209:
15204:
15202:
15199:
15197:
15194:
15193:
15191:
15187:on the whole
15183:
15177:
15174:
15170:
15167:
15166:
15165:
15162:
15160:
15159:type-2 Gumbel
15157:
15155:
15152:
15150:
15147:
15145:
15142:
15140:
15137:
15135:
15132:
15130:
15127:
15125:
15122:
15120:
15117:
15115:
15112:
15110:
15107:
15105:
15102:
15100:
15097:
15095:
15092:
15090:
15087:
15085:
15082:
15080:
15077:
15075:
15072:
15070:
15067:
15065:
15062:
15060:
15057:
15053:
15050:
15049:
15048:
15045:
15043:
15041:
15036:
15034:
15031:
15029:
15028:Half-logistic
15026:
15022:
15019:
15018:
15017:
15014:
15012:
15009:
15005:
15002:
15000:
14997:
14996:
14995:
14992:
14990:
14987:
14985:
14984:Folded normal
14982:
14978:
14975:
14974:
14973:
14972:
14968:
14964:
14961:
14959:
14956:
14954:
14951:
14950:
14949:
14946:
14942:
14939:
14938:
14937:
14934:
14932:
14929:
14927:
14924:
14918:
14915:
14914:
14913:
14910:
14908:
14905:
14904:
14903:
14900:
14898:
14895:
14893:
14890:
14888:
14885:
14883:
14880:
14878:
14875:
14873:
14870:
14869:
14867:
14859:
14853:
14850:
14848:
14845:
14843:
14840:
14838:
14835:
14833:
14830:
14828:
14827:Raised cosine
14825:
14823:
14820:
14818:
14815:
14813:
14810:
14808:
14805:
14803:
14800:
14798:
14795:
14793:
14790:
14788:
14785:
14783:
14780:
14778:
14775:
14773:
14770:
14768:
14765:
14764:
14762:
14756:
14753:
14747:
14737:
14734:
14732:
14729:
14727:
14724:
14722:
14719:
14717:
14714:
14712:
14709:
14707:
14704:
14702:
14701:Mixed Poisson
14699:
14697:
14694:
14692:
14689:
14687:
14684:
14682:
14679:
14677:
14674:
14672:
14669:
14667:
14664:
14662:
14659:
14657:
14654:
14652:
14649:
14648:
14646:
14640:
14634:
14631:
14629:
14626:
14624:
14621:
14619:
14616:
14614:
14611:
14609:
14606:
14602:
14599:
14598:
14597:
14594:
14592:
14589:
14587:
14584:
14582:
14581:Beta-binomial
14579:
14577:
14574:
14572:
14569:
14568:
14566:
14560:
14557:
14551:
14546:
14542:
14535:
14530:
14528:
14523:
14521:
14516:
14515:
14512:
14506:
14503:
14501:
14498:
14493:
14492:
14487:
14484:
14479:
14477:
14474:
14470:
14466:
14465:
14460:
14456:
14455:
14445:
14441:
14438:
14432:
14426:
14422:
14419:
14413:
14405:
14401:
14395:
14386:
14381:
14377:
14373:
14369:
14362:
14354:
14350:
14343:
14325:
14318:
14317:
14309:
14302:
14298:
14295:
14291:
14290:
14285:
14281:
14280:McCullagh, P.
14276:
14274:
14266:
14264:0-8018-6866-1
14260:
14256:
14252:
14248:
14244:
14240:
14236:
14229:
14218:
14214:
14210:
14203:
14196:
14194:
14179:
14175:
14170:
14165:
14161:
14157:
14153:
14146:
14144:
14142:
14133:
14129:
14125:
14121:
14117:
14113:
14106:
14098:
14091:
14080:
14076:
14072:
14068:
14064:
14057:
14050:
14042:
14038:
14034:
14030:
14026:
14022:
14018:
14012:
14010:
14001:
13997:
13993:
13989:
13985:
13981:
13974:
13972:
13970:
13968:
13959:
13953:
13945:
13941:
13937:
13933:
13926:
13924:
13915:
13911:
13907:
13903:
13900:(1): 97–105.
13899:
13895:
13888:
13880:
13876:
13872:
13868:
13864:
13860:
13853:
13838:
13834:
13828:
13813:
13809:
13805:
13801:
13797:
13792:
13787:
13783:
13779:
13775:
13771:
13764:
13757:
13741:
13737:
13733:
13726:
13724:
13709:on 2011-09-30
13705:
13701:
13697:
13693:
13689:
13682:
13675:
13667:
13663:
13659:
13655:
13648:
13640:
13636:
13632:
13628:
13623:
13618:
13614:
13610:
13603:
13594:
13589:
13582:
13566:
13562:
13556:
13548:
13544:
13539:
13534:
13530:
13526:
13525:
13520:
13513:
13505:
13499:
13495:
13491:
13490:
13482:
13474:
13470:
13466:
13462:
13457:
13452:
13448:
13444:
13443:
13435:
13427:
13425:0-471-42798-5
13421:
13417:
13413:
13412:
13404:
13396:
13390:
13386:
13381:
13380:
13371:
13363:
13357:
13353:
13348:
13347:
13338:
13331:
13325:
13319:, Chapter 16.
13316:
13309:
13307:
13305:
13300:
13291:
13288:
13286:
13283:
13281:
13278:
13275:
13272:
13270:
13266:
13263:
13262:
13253:
13249:
13248:value at risk
13245:
13241:
13237:
13234:
13230:
13227:
13223:
13219:
13216:based on the
13215:
13210:
13206:
13202:
13201:
13198:
13194:
13189:
13170:
13167:
13164:
13161:
13156:
13153:
13150:
13146:
13142:
13137:
13132:
13127:
13124:
13119:
13116:
13113:
13109:
13082:
13071:
13067:
13064:
13060:
13056:
13052:
13048:
13044:
13043:
13037:
13035:
13031:
13027:
13023:
13017:
12989:
12986:
12983:
12977:
12971:
12968:
12965:
12952:
12949:
12926:
12923:
12920:
12914:
12911:
12908:
12903:
12898:
12895:
12892:
12889:
12886:
12863:
12859:
12853:
12849:
12845:
12842:
12838:
12834:
12828:
12784:
12781:
12758:
12755:
12752:
12739:
12736:
12728:
12725:
12721:
12719:
12715:
12697:
12694:
12691:
12685:
12682:
12679:
12676:
12653:
12650:
12647:
12644:
12641:
12638:
12635:
12622:
12619:
12611:
12607:
12604:
12600:
12597:
12593:
12578:
12568:
12564:
12560:
12555:
12550:
12546:
12541:
12535:
12526:
12522:
12518:
12513:
12508:
12504:
12497:
12493:
12485:
12481:
12478:
12475:
12469:
12466:
12440:
12437:
12432:
12428:
12421:
12418:
12415:
12412:
12404:
12387:
12384:
12381:
12368:
12362:
12356:
12353:
12330:
12327:
12324:
12318:
12283:
12280:
12272:
12254:
12251:
12248:
12235:
12231:
12226:
12219:
12216:
12210:
12207:
12203:
12199:
12195:
12191:
12188:
12164:
12161:
12158:
12145:
12142:
12134:
12116:
12113:
12110:
12097:
12091:
12088:
12065:
12062:
12059:
12052:
12049:
12046:
12033:
12030:
12027:
12024:
12016:
12014:
12012:
11991:
11988:
11985:
11974:
11971:
11948:
11942:
11939:
11936:
11930:
11927:
11920:
11918:
11916:
11895:
11892:
11871:
11865:
11862:
11859:
11853:
11850:
11843:
11842:
11818:
11814:
11809:
11806:
11794:
11791:
11788:
11784:
11780:
11777:
11766:
11760:
11751:
11747:
11744:
11741:
11733:
11725:
11718:
11717:
11716:
11713:
11697:
11694:
11691:
11687:
11683:
11678:
11675:
11672:
11668:
11647:
11644:
11641:
11619:
11616:
11613:
11609:
11605:
11600:
11597:
11594:
11590:
11566:
11563:
11558:
11551:
11547:
11544:
11541:
11537:
11532:
11523:
11520:
11517:
11507:
11498:
11491:
11488:
11485:
11481:
11477:
11471:
11467:
11464:
11461:
11457:
11452:
11444:
11441:
11438:
11434:
11430:
11423:
11420:
11417:
11413:
11408:
11404:
11398:
11395:
11387:
11375:
11374:
11373:
11355:
11348:
11345:
11337:
11326:
11323:
11318:
11315:
11312:
11308:
11293:
11288:
11284:
11281:
11278:
11274:
11269:
11266:
11263:
11259:
11255:
11251:
11241:
11240:
11239:
11221:
11218:
11215:
11212:
11209:
11203:
11200:
11197:
11194:
11174:
11166:
11162:
11149:
11133:
11112:
11106:
11103:
11100:
11095:
11092:
11089:
11083:
11079:
11076:
11073:
11067:
11064:
11061:
11056:
11053:
11050:
11024:
11018:
11015:
11012:
11009:
11001:
10986:
10966:
10946:
10926:
10905:
10899:
10896:
10893:
10890:
10885:
10882:
10879:
10876:
10870:
10866:
10863:
10860:
10854:
10851:
10848:
10845:
10840:
10837:
10834:
10831:
10805:
10799:
10796:
10793:
10790:
10762:
10754:
10749:
10745:
10738:
10735:
10732:
10729:
10706:
10700:
10697:
10694:
10691:
10671:
10668:
10665:
10660:
10656:
10652:
10649:
10641:
10638:
10618:
10615:
10609:
10606:
10600:
10597:
10594:
10588:
10585:
10559:
10556:
10553:
10547:
10544:
10541:
10538:
10530:
10511:
10507:
10503:
10498:
10494:
10490:
10485:
10481:
10477:
10472:
10468:
10461:
10458:
10455:
10452:
10449:
10446:
10421:
10417:
10413:
10408:
10404:
10400:
10395:
10391:
10387:
10382:
10378:
10371:
10368:
10365:
10362:
10359:
10356:
10331:
10327:
10323:
10318:
10314:
10307:
10304:
10301:
10298:
10273:
10269:
10265:
10260:
10256:
10249:
10246:
10243:
10240:
10232:
10210:
10202:
10199:
10196:
10193:
10190:
10185:
10181:
10167:
10164:
10161:
10158:
10155:
10132:
10129:
10124:
10120:
10113:
10110:
10107:
10104:
10096:
10095:
10089:
10072:
10063:
10059:
10056:
10053:
10046:
10037:
10032:
10017:
10014:
10000:
9990:
9985:
9972:
9969:
9965:
9957:
9954:
9948:
9938:
9930:
9927:
9922:
9913:
9910:
9896:
9891:
9887:
9884:
9881:
9875:
9865:
9859:
9856:
9846:
9841:
9836:
9823:
9816:
9815:
9814:
9800:
9792:
9773:
9769:
9761:
9757:
9753:
9743:
9739:
9735:
9730:
9718:
9714:
9710:
9707:
9695:
9690:
9683:
9680:
9676:
9671:
9665:
9662:
9657:
9653:
9649:
9646:
9640:
9633:
9632:
9631:
9628:
9626:
9610:
9590:
9570:
9550:
9527:
9523:
9515:
9511:
9507:
9497:
9493:
9489:
9484:
9474:
9470:
9466:
9463:
9457:
9452:
9442:
9438:
9434:
9431:
9421:
9416:
9409:
9406:
9402:
9397:
9391:
9388:
9383:
9379:
9375:
9370:
9366:
9362:
9359:
9356:
9353:
9347:
9340:
9339:
9338:
9335:
9321:
9298:
9292:
9286:
9278:
9270:
9264:
9261:
9255:
9248:
9234:
9228:
9220:
9216:
9212:
9209:
9203:
9200:
9192:
9188:
9180:
9179:
9178:
9161:
9155:
9147:
9128:
9120:
9116:
9092:
9086:
9063:
9055:
9051:
9026:
9018:
9012:
9009:
9003:
8995:
8991:
8987:
8983:
8979:
8973:
8965:
8961:
8953:
8952:
8951:
8937:
8932:
8928:
8924:
8921:
8899:
8889:
8886:
8864:
8860:
8854:
8850:
8846:
8843:
8840:
8835:
8831:
8825:
8821:
8817:
8814:
8792:
8782:
8778:
8774:
8771:
8768:
8763:
8759:
8752:
8749:
8742:
8741:random vector
8732:
8718:
8715:
8704:
8693:
8690:
8667:
8664:
8661:
8635:
8632:
8623:
8607:
8603:
8594:
8576:
8572:
8549:
8545:
8522:
8518:
8509:
8493:
8490:
8470:
8448:
8445:
8442:
8439:
8417:
8413:
8389:
8379:
8375:
8371:
8366:
8362:
8350:
8347:
8344:
8334:
8330:
8326:
8321:
8317:
8301:
8300:
8299:
8298:must satisfy
8285:
8265:
8237:
8232:
8226:
8222:
8218:
8213:
8209:
8204:
8199:
8194:
8190:
8183:
8178:
8172:
8168:
8164:
8159:
8155:
8150:
8141:
8136:
8133:
8130:
8126:
8118:
8117:
8116:
8099:
8096:
8091:
8088:
8083:
8072:
8067:
8061:
8057:
8053:
8048:
8044:
8039:
8034:
8029:
8025:
8018:
8011:
8006:
8000:
7996:
7992:
7987:
7983:
7978:
7973:
7965:
7960:
7957:
7954:
7950:
7946:
7940:
7937:
7932:
7929:
7919:
7905:
7902:
7894:
7889:
7883:
7879:
7874:
7869:
7865:
7860:
7855:
7850:
7846:
7835:
7831:
7827:
7822:
7818:
7811:
7803:
7798:
7795:
7792:
7788:
7784:
7776:
7772:
7768:
7763:
7760:
7750:
7749:
7748:
7734:
7712:
7708:
7683:
7677:
7672:
7667:
7661:
7657:
7653:
7648:
7644:
7637:
7632:
7629:
7625:
7621:
7618:
7613:
7608:
7605:
7602:
7598:
7594:
7588:
7585:
7579:
7576:
7573:
7570:
7567:
7561:
7558:
7553:
7549:
7544:
7539:
7535:
7531:
7528:
7525:
7520:
7516:
7503:
7493:
7492:
7491:
7477:
7457:
7435:
7431:
7422:
7418:
7416:
7398:
7394:
7385:
7381:
7365:
7357:
7339:
7335:
7314:
7292:
7288:
7278:
7262:
7258:
7235:
7231:
7208:
7204:
7178:
7174:
7168:
7163:
7160:
7157:
7153:
7147:
7144:
7139:
7130:
7120:
7119:
7118:
7116:
7105:
7103:
7099:
7096:Consider the
7089:
7087:
7082:
7079:
7075:
7055:
7024:
7018:
7015:
7012:
7009:
7006:
6993:
6989:
6985:
6982:
6979:
6970:
6966:
6962:
6959:
6955:
6940:
6936:
6932:
6929:
6926:
6913:
6909:
6905:
6903:
6895:
6892:
6883:
6879:
6875:
6872:
6868:
6863:
6860:
6847:
6843:
6839:
6836:
6833:
6824:
6820:
6816:
6813:
6807:
6803:
6787:
6783:
6779:
6777:
6767:
6763:
6756:
6742:
6741:
6740:
6738:
6714:
6708:
6704:
6700:
6697:
6678:
6674:
6670:
6662:
6652:
6641:
6631:
6630:
6629:
6612:
6609:
6606:
6580:
6577:
6554:
6551:
6548:
6545:
6539:
6536:
6522:
6520:
6516:
6511:
6509:
6508:
6503:
6487:
6484:
6477:
6471:
6468:
6463:
6458:
6455:
6452:
6448:
6436:
6411:
6408:
6401:
6395:
6392:
6387:
6382:
6379:
6375:
6363:
6350:
6345:
6343:
6342:
6337:
6336:
6329:
6315:
6302:
6295:
6293:
6279:
6276:
6269:
6263:
6260:
6250:
6246:
6242:
6239:
6236:
6229:
6223:
6220:
6215:
6207:
6203:
6195:
6194:
6191:
6189:
6180:
6173:
6171:
6157:
6154:
6151:
6144:
6138:
6135:
6122:
6118:
6110:
6109:
6106:
6089:
6083:
6076:
6072:
6062:
6059:
6045:
6042:
6039:
6036:
6031:
6027:
6023:
6018:
6014:
5993:
5990:
5987:
5984:
5979:
5975:
5971:
5966:
5962:
5947:
5943:
5927:
5917:
5913:
5909:
5904:
5900:
5891:
5886:
5883:
5880:
5876:
5870:
5867:
5862:
5857:
5853:
5843:
5827:
5823:
5817:
5812:
5809:
5806:
5802:
5796:
5793:
5788:
5783:
5779:
5758:
5755:
5750:
5746:
5742:
5737:
5733:
5718:
5696:
5693:
5690:
5687:
5676:
5672:
5667:
5661:
5651:
5647:
5643:
5640:
5634:
5631:
5625:
5622:
5616:
5606:
5605:
5604:
5590:
5582:
5560:
5557:
5554:
5548:
5545:
5542:
5539:
5524:
5521:
5518:
5511:
5508:
5500:
5495:
5490:
5486:
5482:
5476:
5470:
5463:
5462:
5461:
5459:
5439:
5435:
5430:
5423:
5420:
5414:
5411:
5407:
5403:
5399:
5393:
5388:
5382:
5378:
5375:
5369:
5366:
5363:
5356:
5353:
5345:
5344:
5343:
5341:
5315:
5312:
5309:
5303:
5300:
5297:
5295:
5287:
5284:
5274:
5271:
5266:
5262:
5258:
5255:
5249:
5243:
5240:
5234:
5231:
5226:
5222:
5218:
5215:
5209:
5196:
5192:
5188:
5185:
5183:
5175:
5169:
5158:
5157:
5156:
5148:
5146:
5142:
5138:
5134:
5115:
5107:
5103:
5097:
5093:
5089:
5082:
5077:
5071:
5068:
5065:
5061:
5057:
5052:
5049:
5046:
5042:
5037:
5032:
5027:
5022:
5016:
5012:
5008:
5003:
4999:
4994:
4986:
4983:
4980:
4976:
4968:
4964:
4960:
4955:
4952:
4949:
4945:
4940:
4936:
4929:
4925:
4921:
4916:
4913:
4910:
4906:
4901:
4896:
4880:
4879:
4878:
4876:
4866:
4864:
4848:
4845:
4842:
4834:
4830:
4810:
4807:
4801:
4798:
4795:
4792:
4788:
4781:
4778:
4773:
4769:
4765:
4762:
4754:
4750:
4736:
4732:
4725:
4722:
4718:
4713:
4707:
4704:
4699:
4695:
4691:
4688:
4682:
4675:
4674:
4673:
4671:
4652:
4642:
4634:
4631:
4628:
4623:
4619:
4615:
4611:
4607:
4604:
4601:
4595:
4592:
4589:
4585:
4578:
4575:
4570:
4566:
4562:
4559:
4553:
4540:
4536:
4532:
4528:
4523:
4520:
4517:
4513:
4509:
4505:
4499:
4493:
4485:
4481:
4473:
4472:
4471:
4469:
4453:
4439:
4425:
4403:
4399:
4393:
4388:
4385:
4382:
4378:
4372:
4369:
4345:
4341:
4338:
4332:
4329:
4326:
4319:
4313:
4308:
4304:
4298:
4293:
4290:
4286:
4280:
4277:
4264:
4239:
4219:
4216:
4211:
4207:
4203:
4198:
4194:
4179:
4177:
4173:
4168:
4152:
4148:
4137:
4133:
4122:
4118:
4095:
4091:
4085:
4081:
4075:
4071:
4048:
4044:
4038:
4034:
4028:
4024:
4001:
3997:
3993:
3990:
3987:
3982:
3978:
3955:
3951:
3947:
3944:
3941:
3936:
3932:
3909:
3905:
3901:
3898:
3895:
3890:
3886:
3863:
3859:
3855:
3852:
3849:
3844:
3840:
3836:
3831:
3827:
3817:
3797:
3767:
3759:
3756:
3752:
3748:
3742:
3732:
3728:
3722:
3718:
3713:
3692:
3682:
3674:
3670:
3666:
3662:
3657:
3654:
3651:
3647:
3643:
3639:
3633:
3627:
3619:
3615:
3606:
3601:
3585:
3581:
3575:
3571:
3565:
3562:
3557:
3548:
3538:
3534:
3516:
3512:
3508:
3505:
3502:
3497:
3493:
3489:
3484:
3480:
3465:
3463:
3459:
3455:
3451:
3447:
3442:
3440:
3436:
3431:
3415:
3411:
3402:
3398:
3395:defined. Its
3394:
3390:
3386:
3362:
3356:
3353:
3350:
3324:
3317:
3313:
3307:
3303:
3295:
3291:
3285:
3280:
3277:
3274:
3270:
3262:
3261:
3260:
3258:
3255:(defining a
3242:
3239:
3236:
3233:
3230:
3227:
3224:
3221:
3218:
3215:
3212:
3207:
3203:
3182:
3179:
3174:
3170:
3166:
3163:
3160:
3155:
3151:
3130:
3110:
3090:
3070:
3044:
3041:
3035:
3032:
3029:
3026:
3023:
3016:
3000:
2997:
2994:
2965:
2963:
2962:-distribution
2961:
2933:
2930:
2925:
2921:
2918:
2915:
2911:
2908:
2903:
2900:
2895:
2889:
2886:
2883:
2880:
2877:
2871:
2864:
2863:
2862:
2845:
2839:
2836:
2829:
2824:
2810:
2807:
2799:
2780:
2777:
2772:
2768:
2764:
2761:
2758:
2753:
2749:
2722:
2718:
2713:
2706:
2703:
2697:
2694:
2690:
2686:
2682:
2678:
2675:
2671:
2668:
2663:
2659:
2655:
2649:
2646:
2641:
2637:
2633:
2630:
2624:
2617:
2616:
2615:
2613:
2609:
2588:
2585:
2580:
2576:
2571:
2565:
2561:
2557:
2554:
2548:
2544:
2541:
2536:
2533:
2528:
2522:
2519:
2514:
2510:
2506:
2503:
2497:
2490:
2489:
2488:
2486:
2476:
2462:
2456:
2453:
2449:
2444:
2441:
2421:
2398:
2394:
2386:
2382:
2378:
2373:
2363:
2359:
2355:
2352:
2343:
2339:
2333:
2329:
2326:
2320:
2314:
2309:
2304:
2298:
2294:
2290:
2287:
2281:
2276:
2273:
2269:
2265:
2260:
2254:
2251:
2248:
2245:
2240:
2236:
2232:
2229:
2223:
2216:
2215:
2214:
2196:
2185:
2181:
2177:
2174:
2168:
2164:
2159:
2153:
2150:
2147:
2144:
2141:
2135:
2128:
2127:
2126:
2124:
2108:
2105:
2102:
2082:
2079:
2074:
2070:
2045:
2039:
2036:
2033:
2028:
2025:
2019:
2005:
2002:
1997:
1993:
1987:
1984:
1981:
1977:
1972:
1958:
1955:
1950:
1944:
1941:
1938:
1932:
1925:
1924:
1923:
1910:
1907:
1904:
1899:
1895:
1891:
1888:
1879:
1863:
1859:
1855:
1852:
1845:, located at
1829:
1826:
1822:
1806:
1804:
1800:
1799:infinitesimal
1796:
1792:
1788:
1772:
1764:
1748:
1745:
1737:
1721:
1713:
1695:
1691:
1667:
1663:
1655:
1651:
1647:
1642:
1632:
1628:
1624:
1621:
1614:
1609:
1603:
1600:
1595:
1588:
1582:
1577:
1572:
1566:
1562:
1558:
1555:
1549:
1544:
1541:
1537:
1533:
1530:
1526:
1521:
1515:
1512:
1507:
1503:
1499:
1496:
1490:
1483:
1482:
1481:
1479:
1469:
1455:
1451:
1447:
1424:
1421:
1418:
1406:
1392:
1388:
1384:
1376:
1372:
1368:
1365:
1349:
1329:
1305:
1296:
1293:
1288:
1285:
1279:
1275:
1271:
1268:
1265:
1262:
1255:
1254:
1253:
1239:
1216:
1213:
1210:
1187:
1178:
1164:
1142:
1138:
1117:
1094:
1091:
1086:
1082:
1069:
1060:
1055:Constructions
1052:
1050:
1046:
1042:
1037:
1033:
1029:
1025:
1017:
1012:
1003:
1001:
997:
993:
988:
986:
982:
978:
974:
970:
965:
963:
959:
955:
951:
947:
943:
938:
936:
932:
913:
910:
905:
901:
870:
867:
862:
858:
854:
851:
845:
837:
833:
829:
825:
821:
817:
813:
809:
805:
779:
775:
771:
767:
756:
752:
727:
718:
715:
712:
708:
702:
698:
691:
688:
678:
674:
668:
664:
645:
642:
639:
633:
630:
621:
617:
614:
609:
605:
602:
597:
593:
578:
569:
565:
562:
557:
553:
535:
531:
521:
517:
499:
495:
485:
481:
478:
473:
469:
445:
442:
436:
433:
427:
421:
418:
414:
411:
406:
402:
392:
388:
370:
367:
362:
358:
353:
347:
343:
339:
336:
330:
326:
323:
318:
315:
304:
300:
280:
274:
269:
264:
258:
254:
250:
247:
241:
236:
233:
229:
224:
221:
217:
206:
202:
179:
176:
170:
164:
161:
151:
147:
143:
128:
125:
122:
113:
109:
91:
87:
77:
73:
69:
61:
53:
45:
37:
36:Lorenz system
33:
19:
15786:
15774:
15740:Multivariate
15739:
15727:
15715:
15710:Wrapped Lévy
15670:
15618:Matrix gamma
15611:
15591:
15579:Normal-gamma
15572:
15538:Continuous:
15537:
15508:
15453:Tukey lambda
15440:
15432:
15427:-exponential
15424:
15416:
15407:
15398:
15389:
15383:-exponential
15380:
15324:
15291:
15258:
15220:
15207:
15195:
15134:Poly-Weibull
15079:Log-logistic
15039:
15038:Hotelling's
14970:
14812:Logit-normal
14686:Gauss–Kuzmin
14681:Flory–Schulz
14562:with finite
14489:
14462:
14431:
14412:
14394:
14375:
14371:
14361:
14348:
14342:
14331:. Retrieved
14315:
14308:
14287:
14238:
14234:
14228:
14217:the original
14212:
14208:
14181:. Retrieved
14159:
14155:
14115:
14111:
14105:
14096:
14090:
14079:the original
14066:
14062:
14049:
14024:
14020:
13983:
13979:
13952:cite journal
13935:
13931:
13897:
13893:
13887:
13862:
13858:
13852:
13841:. Retrieved
13827:
13816:, retrieved
13773:
13769:
13756:
13744:. Retrieved
13735:
13711:. Retrieved
13704:the original
13691:
13687:
13674:
13660:(1): 54–59.
13657:
13653:
13647:
13612:
13608:
13602:
13581:
13569:. Retrieved
13564:
13555:
13528:
13522:
13512:
13488:
13481:
13446:
13440:
13434:
13410:
13403:
13378:
13370:
13345:
13337:
13329:
13324:
13314:
13269:Lévy process
13047:spectroscopy
13019:
12013:distribution
12010:
11917:distribution
11914:
11714:
11581:
11371:
11158:
11155:Lévy measure
10087:
9788:
9629:
9542:
9336:
9313:
9042:
8738:
8624:
8404:
8257:
8114:
7699:
7419:
7279:
7195:
7114:
7111:
7095:
7083:
7043:
6734:
6528:
6512:
6505:
6501:
6346:
6339:
6333:
6330:
6307:
6296:
6185:
6174:
6068:
6060:
5953:
5844:
5724:
5716:
5578:
5455:
5337:
5154:
5133:f-divergence
5130:
4872:
4832:
4828:
4826:
4667:
4445:
4185:
4169:
3818:
3602:
3471:
3443:
3432:
3382:
2966:
2959:
2955:
2825:
2737:
2605:
2482:
2413:
2212:
2122:
2061:
1880:
1812:
1682:
1475:
1407:
1321:
1179:
1070:
1066:
1058:
1021:
989:
966:
942:pathological
939:
835:
831:
827:
819:
803:
801:
32:Lorenz curve
15824:Exponential
15673:directional
15662:Directional
15549:Generalized
15520:Multinomial
15475:continuous-
15415:Kaniadakis
15406:Kaniadakis
15397:Kaniadakis
15388:Kaniadakis
15379:Kaniadakis
15331:Tracy–Widom
15308:Skew normal
15290:Noncentral
15074:Log-Laplace
15052:Generalized
15033:Half-normal
14999:Generalized
14963:Logarithmic
14948:Exponential
14902:Chi-squared
14842:U-quadratic
14807:Kumaraswamy
14749:Continuous
14696:Logarithmic
14591:Categorical
14099:. Elsevier.
14069:(6): 1901.
13265:Lévy flight
13195:, see also
6737:raw moments
3924:and scales
3537:sample mean
969:mathematics
15923:Power laws
15902:Categories
15819:Elliptical
15775:Degenerate
15761:Degenerate
15509:Discrete:
15468:univariate
15323:Student's
15278:Asymmetric
15257:Johnson's
15185:supported
15129:Phase-type
15084:Log-normal
15069:Log-Cauchy
15059:Kolmogorov
14977:Noncentral
14907:Noncentral
14887:Beta prime
14837:Triangular
14832:Reciprocal
14802:Irwin–Hall
14751:univariate
14731:Yule–Simon
14613:Rademacher
14555:univariate
14333:2016-05-04
14289:Biometrika
14183:2017-01-07
13843:2014-11-22
13818:2019-09-25
13713:2011-06-02
13622:2101.12459
13593:1905.10965
13456:1505.01957
13296:References
12605:of type 7.
11913:Student's
8115:Note that
5603:for which
4252:such that
3391:or higher
3379:Properties
3143:such that
2958:Student's
1157:and scale
816:physicists
76:Parameters
15544:Dirichlet
15525:Dirichlet
15435:-Gaussian
15410:-Logistic
15247:Holtsmark
15219:Gaussian
15206:Fisher's
15189:real line
14691:Geometric
14671:Delaporte
14576:Bernoulli
14553:Discrete
14491:MathWorld
14469:EMS Press
13914:123586208
13639:231728407
13240:fat tails
13209:hydrology
13165:β
13147:ε
13128:β
13086:^
13083:β
13030:resonance
12987:≥
12953:∼
12921:μ
12915:
12909:∼
12893:μ
12829:
12785:∼
12740:∼
12698:γ
12692:μ
12686:
12680:∼
12654:μ
12648:γ
12623:∼
12598:of type 4
12565:γ
12542:γ
12523:γ
12482:
12476:∼
12441:γ
12422:
12416:∼
12369:∼
12357:
12319:
12284:∼
12236:∼
12211:−
12200:π
12192:
12146:∼
12098:∼
12034:∼
11992:σ
11986:μ
11949:∼
11943:σ
11937:μ
11931:
11872:∼
11854:
11781:−
11761:∖
11752:∫
11748:
11726:π
11698:γ
11679:γ
11642:γ
11620:γ
11601:γ
11524:γ
11492:γ
11445:γ
11424:γ
11388:γ
11384:Π
11338:γ
11334:Π
11324:−
11294:∫
11285:
11252:
11210:γ
11204:
11198:∼
11175:γ
11101:ψ
11093:−
11090:ψ
11080:
11074:∼
11054:−
11025:ψ
11019:
11013:∼
10894:ψ
10880:ψ
10867:
10861:∼
10806:ψ
10800:
10794:∼
10763:γ
10739:
10733:∼
10707:ψ
10701:
10695:∼
10684:, define
10672:γ
10650:ψ
10619:γ
10601:
10595:∼
10560:γ
10548:
10542:∼
10508:γ
10495:γ
10478:−
10462:
10456:∼
10450:−
10418:γ
10405:γ
10372:
10366:∼
10328:γ
10308:
10302:∼
10270:γ
10250:
10244:∼
10203:γ
10197:ℓ
10168:∼
10165:ℓ
10133:γ
10114:
10108:∼
10038:μ
10033:−
10015:−
10009:Σ
9991:μ
9986:−
9944:Σ
9923:π
9903:Γ
9872:Γ
9852:Σ
9842:μ
9740:γ
9711:−
9696:γ
9684:π
9666:γ
9494:γ
9467:−
9435:−
9422:γ
9410:π
9392:γ
9287:γ
9256:γ
9156:γ
9087:γ
9013:γ
9010:−
8962:φ
8890:∈
8844:⋯
8772:…
8719:γ
8694:
8668:γ
8636:∼
8508:numerical
8471:γ
8446:−
8372:−
8351:≤
8348:γ
8345:≤
8327:−
8286:γ
8266:γ
8219:−
8191:γ
8165:−
8127:∑
8092:γ
8084:−
8054:−
8026:γ
8019:γ
7993:−
7951:∑
7941:γ
7933:ℓ
7875:−
7847:γ
7828:−
7789:∑
7764:ℓ
7735:γ
7668:γ
7654:−
7622:
7599:∑
7595:−
7589:π
7586:γ
7580:
7571:−
7562:γ
7545:∣
7529:…
7507:^
7504:ℓ
7458:γ
7415:fat tails
7366:γ
7315:γ
7154:∑
7134:¯
7059:∞
7056:−
7053:∞
7022:∞
7016:π
7013:−
7002:∞
6997:∞
6994:−
6990:∫
6949:∞
6944:∞
6941:−
6937:∫
6933:−
6922:∞
6917:∞
6914:−
6910:∫
6864:−
6856:∞
6851:∞
6848:−
6844:∫
6796:∞
6791:∞
6788:−
6784:∫
6780:∝
6757:
6698:π
6675:γ
6642:
6613:γ
6581:∼
6546:−
6540:∈
6453:−
6449:∫
6443:∞
6440:→
6380:−
6376:∫
6370:∞
6367:→
6256:∞
6247:∫
6211:∞
6208:−
6204:∫
6131:∞
6126:∞
6123:−
6119:∫
5950:infinity.
5910:−
5877:∑
5803:∑
5759:…
5694:
5673:γ
5644:−
5626:
5617:
5561:γ
5558:π
5549:
5525:γ
5487:∫
5477:γ
5415:−
5404:π
5383:π
5379:γ
5370:γ
5316:γ
5313:π
5304:
5275:γ
5244:
5235:γ
5205:∞
5200:∞
5197:−
5193:∫
5189:−
5176:γ
5104:γ
5094:γ
5058:−
5013:γ
5000:γ
4987:
4965:γ
4926:γ
4793:−
4782:γ
4751:φ
4745:∞
4740:∞
4737:−
4733:∫
4726:π
4708:γ
4635:γ
4632:−
4579:γ
4549:∞
4544:∞
4541:−
4537:∫
4506:
4482:φ
4426:γ
4379:∑
4346:π
4342:γ
4314:ρ
4291:−
4287:∫
4271:∞
4268:→
4240:ρ
4220:…
4149:γ
4119:∑
4110:and scale
4072:∑
4025:∑
3991:…
3952:γ
3945:…
3933:γ
3899:…
3853:…
3801:¯
3787:, and so
3757:−
3719:∑
3714:φ
3675:−
3640:
3616:φ
3572:∑
3552:¯
3506:…
3325:∼
3271:∑
3234:…
3213:≥
3164:⋯
3048:Σ
3033:∼
2998:×
2975:Σ
2912:
2904:π
2840:
2811:γ
2781:γ
2762:γ
2759:−
2698:−
2687:π
2679:
2672:γ
2650:γ
2610:(inverse
2572:γ
2558:−
2545:
2537:π
2523:γ
2457:γ
2454:π
2383:γ
2356:−
2340:γ
2305:γ
2291:−
2249:γ
2169:π
2103:γ
2040:ψ
2037:−
2026:−
2006:π
1988:ψ
1985:−
1959:π
1945:ψ
1911:γ
1889:ψ
1830:γ
1827:π
1773:γ
1749:γ
1722:γ
1652:γ
1625:−
1615:γ
1604:π
1573:γ
1559:−
1534:γ
1531:π
1516:γ
1289:−
1280:π
1272:
1165:γ
1095:γ
914:γ
871:γ
818:, as the
776:γ
719:γ
716:−
692:
646:γ
643:π
634:
613:undefined
601:undefined
579:γ
561:undefined
477:undefined
437:−
428:π
422:
415:γ
354:γ
340:−
327:
319:π
265:γ
251:−
225:γ
222:π
183:∞
174:∞
171:−
165:∈
123:γ
15877:Category
15809:Circular
15802:Families
15787:Singular
15766:singular
15530:Negative
15477:discrete
15443:-Weibull
15401:-Weibull
15285:Logistic
15169:Discrete
15139:Rayleigh
15119:Nakagami
15042:-squared
15016:Gompertz
14865:interval
14601:Negative
14586:Binomial
14440:Archived
14421:Archived
14404:Archived
14324:Archived
14297:Archived
13837:Archived
13812:archived
13808:53117661
13740:Archived
13473:31582370
13259:See also
13204:physics.
10722:to mean
9623:are not
9314:for all
6628:we have
5512:′
5357:′
3389:variance
3083:-vector
2606:and the
1765:(FWHM).
1375:variance
1049:Bienaymé
1030:, after
998:and the
979:for the
950:variance
948:and its
596:Skewness
556:Variance
391:Quantile
108:location
15887:Commons
15859:Wrapped
15854:Tweedie
15849:Pearson
15844:Mixture
15751:Bingham
15650:Complex
15640:Inverse
15630:Wishart
15623:Inverse
15610:Matrix
15584:Inverse
15500:(joint)
15419:-Erlang
15273:Laplace
15164:Weibull
15021:Shifted
15004:Inverse
14989:Fréchet
14912:Inverse
14847:Uniform
14767:Arcsine
14726:Skellam
14721:Poisson
14644:support
14618:Soliton
14571:Benford
14564:support
14471:, 2001
14243:Bibcode
14178:2237984
14132:2283210
14041:2286549
14000:2282794
13879:2285535
13800:3315772
13571:21 June
13547:2041858
13193:CumFreq
13022:nuclear
12879:, then
12669:, then
11146:is the
11134:CCauchy
11077:CCauchy
5713:Moments
5151:Entropy
3531:are an
3393:moments
2487:(CDF):
1734:is the
1710:is the
1041:Laplace
1036:Poisson
1006:History
983:in the
958:moments
822:(after
810:, is a
620:Entropy
150:Support
15793:Cantor
15635:Normal
15466:Mixed
15392:-Gamma
15318:Stable
15268:Landau
15242:Gumbel
15196:Cauchy
15124:Pareto
14936:Erlang
14917:Scaled
14872:Benini
14711:Panjer
14349:Optics
14261:
14176:
14130:
14039:
13998:
13912:
13877:
13806:
13798:
13746:5 July
13736:Random
13637:
13545:
13500:
13471:
13422:
13391:
13358:
12912:Cauchy
12683:Cauchy
12628:Stable
12479:Cauchy
12419:Cauchy
12374:Cauchy
12241:Cauchy
12103:Cauchy
11928:Cauchy
11851:Cauchy
11372:where
11201:Stable
11126:where
11040:then:
11016:Cauchy
10919:where
10864:Cauchy
10821:then:
10797:Cauchy
10736:Cauchy
10698:Cauchy
10598:Cauchy
10545:Cauchy
10459:Cauchy
10369:Cauchy
10305:Cauchy
10247:Cauchy
10173:Cauchy
10111:Cauchy
9583:is 0,
9043:where
8691:median
8683:, the
7102:i.i.d.
6073:has a
3970:, and
3439:stable
3401:median
2909:arctan
2837:arctan
2542:arctan
2414:where
1683:where
1480:(PDF)
1373:0 and
1032:Agnesi
1024:Fermat
484:Median
324:arctan
144:(real)
42:Cauchy
15515:Ewens
15341:Voigt
15313:Slash
15094:Lomax
15089:Log-t
14994:Gamma
14941:Hyper
14931:Davis
14926:Dagum
14782:Bates
14772:ARGUS
14656:Borel
14327:(PDF)
14320:(PDF)
14220:(PDF)
14205:(PDF)
14174:JSTOR
14128:JSTOR
14082:(PDF)
14059:(PDF)
14037:JSTOR
13996:JSTOR
13910:S2CID
13875:JSTOR
13804:S2CID
13796:JSTOR
13766:(PDF)
13707:(PDF)
13684:(PDF)
13635:S2CID
13617:arXiv
13588:arXiv
13543:JSTOR
13469:S2CID
13451:arXiv
12612:: if
12456:then
12346:then
12181:then
10783:. If
10575:then
10148:then
6069:If a
3452:with
2987:is a
1369:with
1322:When
931:ratio
834:, or
142:scale
15764:and
15722:Kent
15149:Rice
15064:Lévy
14892:Burr
14822:PERT
14787:Beta
14736:Zeta
14628:Zipf
14545:list
14259:ISBN
13958:link
13748:2021
13573:2023
13498:ISBN
13420:ISBN
13389:ISBN
13385:1333
13356:ISBN
13267:and
13168:>
13024:and
12722:The
11582:and
11545:<
11465:>
11238:by:
10979:and
10439:and
10291:and
9603:and
9563:and
9079:and
7727:and
7490:is:
7450:and
7076:and
6065:Mean
5456:The
5131:Any
4873:The
4827:The
4446:Let
3454:real
3399:and
3397:mode
3385:mean
3195:and
3123:and
2095:and
1342:and
802:The
520:Mode
472:Mean
126:>
112:real
15600:LKJ
14897:Chi
14380:doi
14294:PDF
14251:doi
14164:doi
14120:doi
14071:doi
14067:137
14029:doi
13988:doi
13940:doi
13902:doi
13867:doi
13786:hdl
13778:doi
13696:doi
13692:150
13662:doi
13627:doi
13533:doi
13461:doi
13416:305
13352:704
13242:in
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13061:.
13045:In
13020:In
12729:If
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12189:tan
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11282:exp
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10097:If
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8309:min
7619:log
7577:log
6502:not
6500:is
6433:lim
6360:lim
5691:log
5623:log
5546:log
5501:log
5389:sec
5301:log
5241:log
4984:log
4261:lim
4186:If
3533:IID
3472:If
2967:If
2800:is
2676:tan
2612:cdf
1761:is
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967:In
826:),
689:exp
667:MGF
631:log
568:MAD
419:tan
303:CDF
205:PDF
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14467:,
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13303:^
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12354:ln
11745:PV
11712:.
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9627:.
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1967:Im
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1793:.
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1002:.
987:.
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830:,
677:CF
15612:t
15573:t
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15433:q
15425:q
15417:κ
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15399:κ
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12467:1
12444:)
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5985:,
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5828:i
5824:X
5818:n
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5789:=
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5780:S
5756:,
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5747:X
5743:,
5738:1
5734:X
5697:4
5688:=
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5668:/
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5638:(
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5591:X
5564:)
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5543:=
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5522:;
5519:p
5516:(
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4956:2
4953:,
4950:0
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4937::
4930:1
4922:,
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4914:,
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4808:d
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4705:,
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4629:t
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4571:0
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4317:(
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4217:,
4212:2
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4092:x
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4082:a
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4045:X
4039:i
4035:a
4029:i
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3994:,
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3983:1
3979:a
3956:n
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3942:,
3937:1
3910:n
3906:x
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3891:1
3887:x
3864:n
3860:X
3856:,
3850:,
3845:2
3841:X
3837:,
3832:1
3828:X
3798:X
3772:|
3768:t
3764:|
3760:n
3753:e
3749:=
3746:)
3743:t
3740:(
3733:i
3729:X
3723:i
3693:.
3687:|
3683:t
3679:|
3671:e
3667:=
3663:]
3658:t
3655:X
3652:i
3648:e
3644:[
3637:E
3634:=
3631:)
3628:t
3625:(
3620:X
3586:i
3582:X
3576:i
3566:n
3563:1
3558:=
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3517:n
3513:X
3509:,
3503:,
3498:2
3494:X
3490:,
3485:1
3481:X
3416:0
3412:x
3363:.
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3357:1
3354:,
3351:0
3348:(
3344:y
3341:h
3338:c
3335:u
3332:a
3329:C
3318:j
3314:Y
3308:j
3304:X
3296:j
3292:w
3286:p
3281:1
3278:=
3275:j
3243:,
3240:p
3237:,
3231:,
3228:1
3225:=
3222:i
3219:,
3216:0
3208:i
3204:w
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3171:w
3167:+
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3156:1
3152:w
3131:Y
3111:X
3091:w
3071:p
3051:)
3045:,
3042:0
3039:(
3036:N
3030:Y
3027:,
3024:X
3001:p
2995:p
2960:t
2934:2
2931:1
2926:+
2922:)
2919:x
2916:(
2901:1
2896:=
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2890:1
2887:,
2884:0
2881:;
2878:x
2875:(
2872:F
2849:)
2846:x
2843:(
2808:2
2784:)
2778:+
2773:0
2769:x
2765:,
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2750:x
2746:(
2723:.
2719:]
2714:)
2707:2
2704:1
2695:p
2691:(
2683:[
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2664:0
2660:x
2656:=
2653:)
2647:,
2642:0
2638:x
2634:;
2631:p
2628:(
2625:Q
2589:2
2586:1
2581:+
2577:)
2566:0
2562:x
2555:x
2549:(
2534:1
2529:=
2526:)
2520:,
2515:0
2511:x
2507:;
2504:x
2501:(
2498:F
2463:.
2450:1
2445:=
2442:I
2422:I
2399:,
2395:]
2387:2
2379:+
2374:2
2370:)
2364:0
2360:x
2353:x
2350:(
2344:2
2334:[
2330:I
2327:=
2321:]
2315:2
2310:)
2299:0
2295:x
2288:x
2282:(
2277:+
2274:1
2270:[
2266:I
2261:=
2258:)
2255:I
2252:,
2246:,
2241:0
2237:x
2233:;
2230:x
2227:(
2224:f
2197:.
2191:)
2186:2
2182:x
2178:+
2175:1
2172:(
2165:1
2160:=
2157:)
2154:1
2151:,
2148:0
2145:;
2142:x
2139:(
2136:f
2109:1
2106:=
2083:0
2080:=
2075:0
2071:x
2046:)
2034:x
2029:i
2020:(
2003:1
1998:=
1994:)
1982:x
1978:1
1973:(
1956:1
1951:=
1948:)
1942:;
1939:x
1936:(
1933:f
1908:i
1905:+
1900:0
1896:x
1892:=
1864:0
1860:x
1856:=
1853:x
1823:1
1746:2
1696:0
1692:x
1668:,
1664:]
1656:2
1648:+
1643:2
1639:)
1633:0
1629:x
1622:x
1619:(
1610:[
1601:1
1596:=
1589:]
1583:2
1578:)
1567:0
1563:x
1556:x
1550:(
1545:+
1542:1
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