795:
13258:
8335:
3179:
57:
3191:
3199:
3167:
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1813:
10880:
6148:
13244:
10784:
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13270:
10794:
7384:
7178:
768:
7679:
The concept of the probability distribution and the random variables which they describe underlies the mathematical discipline of probability theory, and the science of statistics. There is spread or variability in almost any value that can be measured in a population (e.g. height of people,
790:
However, for the same use case, it is possible to meet quality control requirements such as that a package of "500 g" of ham must weigh between 490 g and 510 g with at least 98% probability. This is possible because this measurement does not require as much precision from the
615:
6326:. When this phenomenon is studied, the observed states from the subset are as indicated in red. So one could ask what is the probability of observing a state in a certain position of the red subset; if such a probability exists, it is called the probability measure of the system.
4350:
974:
A probability distribution can be described in various forms, such as by a probability mass function or a cumulative distribution function. One of the most general descriptions, which applies for absolutely continuous and discrete variables, is by means of a probability function
3647:âthat is, its cdf increases only where it "jumps" to a higher value, and is constant in intervals without jumps. The points where jumps occur are precisely the values which the random variable may take. Thus the cumulative distribution function has the form
7706:
All of the univariate distributions below are singly peaked; that is, it is assumed that the values cluster around a single point. In practice, actually observed quantities may cluster around multiple values. Such quantities can be modeled using a
8320:) of which some can be fitted more closely to the observed frequency of the data than others, depending on the characteristics of the phenomenon and of the distribution. The distribution giving a close fit is supposed to lead to good predictions.
2480:
In the special case of a real-valued random variable, the probability distribution can equivalently be represented by a cumulative distribution function instead of a probability measure. The cumulative distribution function of a random variable
6576:
does not converge. Formally, the measure exists only if the limit of the relative frequency converges when the system is observed into the infinite future. The branch of dynamical systems that studies the existence of a probability measure is
5044:
is a probability distribution on the real numbers with uncountably many possible values, such as a whole interval in the real line, and where the probability of any event can be expressed as an integral. More precisely, a real random variable
4712:
6793:
1196:), it is more common to study probability distributions whose argument are subsets of these particular kinds of sets (number sets), and all probability distributions discussed in this article are of this type. It is common to denote as
1734:
are applicable to scenarios where the set of possible outcomes can take on values in a continuous range (e.g. real numbers), such as the temperature on a given day. In the absolutely continuous case, probabilities are described by a
4578:
7379:{\displaystyle {\begin{aligned}F(x)=u&\Leftrightarrow 1-e^{-\lambda x}=u\\&\Leftrightarrow e^{-\lambda x}=1-u\\&\Leftrightarrow -\lambda x=\ln(1-u)\\&\Leftrightarrow x={\frac {-1}{\lambda }}\ln(1-u)\end{aligned}}}
4192:
4003:
6653:
are then transformed via some algorithm to create a new random variate having the required probability distribution. With this source of uniform pseudo-randomness, realizations of any random variable can be generated.
7183:
6913:
4187:
4074:
7631:
6584:
Note that even in these cases, the probability distribution, if it exists, might still be termed "absolutely continuous" or "discrete" depending on whether the support is uncountable or countable, respectively.
8232:
7467:
437:
Probability distributions can be defined in different ways and for discrete or for continuous variables. Distributions with special properties or for especially important applications are given specific names.
5324:
4880:
3317:
1824:, also called Gaussian or "bell curve", the most important absolutely continuous random distribution. As notated on the figure, the probabilities of intervals of values correspond to the area under the curve.
3729:
1508:
5673:
1397:
1337:
7115:
5958:
873:
787:
must be zero because no matter how high the level of precision chosen, it cannot be assumed that there are no non-zero decimal digits in the remaining omitted digits ignored by the precision level.
1011:
8297:
or simply distribution fitting is the fitting of a probability distribution to a series of data concerning the repeated measurement of a variable phenomenon. The aim of distribution fitting is to
5862:
6276:
6137:
6084:
5905:
1726:(e.g. a coin toss, a roll of a die) and the probabilities are encoded by a discrete list of the probabilities of the outcomes; in this case the discrete probability distribution is known as
2914:
779:
For example, consider measuring the weight of a piece of ham in the supermarket, and assume the scale can provide arbitrarily many digits of precision. Then, the probability that it weighs
7680:
durability of a metal, sales growth, traffic flow, etc.); almost all measurements are made with some intrinsic error; in physics, many processes are described probabilistically, from the
6307:
or similar. In these cases, the probability distribution is supported on the image of such curve, and is likely to be determined empirically, rather than finding a closed formula for it.
5526:
3160:
4591:
2874:
2748:
2354:: for a discrete random variable, the value with highest probability; for an absolutely continuous random variable, a location at which the probability density function has a local peak.
2799:
7928:, for the distribution of vector magnitudes with Gaussian distributed orthogonal components. Rayleigh distributions are found in RF signals with Gaussian real and imaginary components.
6712:
5745:, which are those having a continuous cumulative distribution function. Every absolutely continuous distribution is a continuous distribution but the inverse is not true, there exist
3482:
6384:
5159:
1089:
7171:
6574:
3594:
1589:
606:
1713:
5113:
4780:
4446:
6997:
6020:
3794:
6305:
6215:
6186:
2690:
2572:
2451:: a property of some distributions in which the portion of the distribution to the left of a specific value (usually the median) is a mirror image of the portion to its right.
3069:
2179:: set of values that can be assumed with non-zero probability (or probability density in the case of a continuous distribution) by the random variable. For a random variable
10830:
4453:
1633:
1038:
5400:
1194:
1172:
935:
477:
6707:
5024:
3875:
3431:
3378:
2421:
2320:
1782:â a list of two or more random variables â taking on various combinations of values. Important and commonly encountered univariate probability distributions include the
1659:
3526:
1900:
1541:
1229:
13313:
7651:
612:
is then defined to be the sum of the probabilities of all outcomes that satisfy the event; for example, the probability of the event "the die rolls an even value" is
6548:
3814:
3767:
3433:. In the case where the range of values is countably infinite, these values have to decline to zero fast enough for the probabilities to add up to 1. For example, if
7699:
The following is a list of some of the most common probability distributions, grouped by the type of process that they are related to. For a more complete list, see
7522:
2282:
2256:
964:
4909:
2224:
900:
553:
3920:
2988:
2641:
2608:
4373:
1923:
7542:
7487:
7037:
7017:
6962:
6939:
6675:
6651:
6621:
6424:
6404:
6040:
5984:
5815:
5733:
5713:
5693:
5583:
5456:
5368:
5348:
5239:
5219:
5199:
5179:
5063:
4981:
4961:
4929:
4800:
4734:
4400:
4106:
3915:
3895:
3834:
3337:
3232:
3008:
2519:
2499:
2380:
2197:
2063:
2043:
2023:
2003:
1975:
1955:
1679:
1609:
1417:
1269:
1249:
1132:
1112:
6516:
6470:
5436:
763:{\displaystyle \ p({\text{â}}2{\text{â}})+p({\text{â}}4{\text{â}})+p({\text{â}}6{\text{â}})={\tfrac {1}{6}}+{\tfrac {1}{6}}+{\tfrac {1}{6}}={\tfrac {1}{2}}~.}
6798:
4111:
4008:
8265:
explains the uncertainties of input variables as probability distribution and provides the power flow calculation also in term of probability distribution.
9442:
5550:
17:
8118:
In quantum mechanics, the probability density of finding the particle at a given point is proportional to the square of the magnitude of the particle's
5244:
4807:
3241:
348:
3650:
10823:
5330:, so that absolutely continuous probability distributions are exactly those with a probability density function. In particular, the probability for
2916:
that satisfies the first four of the properties above is the cumulative distribution function of some probability distribution on the real numbers.
2346:: the value such that the set of values less than the median, and the set greater than the median, each have probabilities no greater than one-half.
772:
In contrast, when a random variable takes values from a continuum then by convention, any individual outcome is assigned probability zero. For such
9345:
8994:
5588:
8588:
8378:
3643:
A real-valued discrete random variable can equivalently be defined as a random variable whose cumulative distribution function increases only by
7042:
1801:
Besides the probability function, the cumulative distribution function, the probability mass function and the probability density function, the
1758:
A probability distribution whose sample space is one-dimensional (for example real numbers, list of labels, ordered labels or binary) is called
12379:
9324:
9236:
8000:
2928:
7811:, for binomial-type observations but where the quantity of interest is the number of failures before the first success; a special case of the
12884:
6217:
are extremely useful to model a myriad of phenomena, since most practical distributions are supported on relatively simple subsets, such as
10945:
10816:
7547:
1806:
4345:{\displaystyle P(X\in E)=\int _{E}f(x)\,dx=\sum _{\omega \in A}p(\omega )\int _{E}\delta (x-\omega )=\sum _{\omega \in A\cap E}p(\omega )}
3733:
The points where the cdf jumps always form a countable set; this may be any countable set and thus may even be dense in the real numbers.
13034:
9571:
7834:, for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed number of total occurrences, sampling using a
7389:
6406:
a subset of the support; if the probability measure exists for the system, one would expect the frequency of observing states inside set
798:
Figure 1: The left graph shows a probability density function. The right graph shows the cumulative distribution function. The value at
8129:
12658:
11299:
10797:
10054:
9138:
Dekking, Frederik Michel; Kraaikamp, Cornelis; LopuhaÀ, Hendrik Paul; Meester, Ludolf Erwin (2005), "Why probability and statistics?",
7805:, for binomial-type observations but where the quantity of interest is the number of failures before a given number of successes occurs
1423:
9962:
5749:, which are neither absolutely continuous nor discrete nor a mixture of those, and do not have a density. An example is given by the
5753:. Some authors however use the term "continuous distribution" to denote all distributions whose cumulative distribution function is
12432:
10749:
7876:
5461:
1344:
1284:
12871:
10615:
9827:
9586:
9435:
3380:. Thus the discrete random variables (i.e. random variables whose probability distribution is discrete) are exactly those with a
5910:
4939:
A special case is the discrete distribution of a random variable that can take on only one fixed value; in other words, it is a
10510:
10274:
341:
7825:, for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed number of total occurrences, using
1809:
also serve to identify a probability distribution, as they uniquely determine an underlying cumulative distribution function.
833:
9948:
9300:
9267:
9212:
9155:
8564:
8501:
8465:
8432:
8115:
to assign probabilities to the occurrence of particular words and word sequences do so by means of probability distributions.
2232:: the regions close to the bounds of the random variable, if the pmf or pdf are relatively low therein. Usually has the form
1743:
is a commonly encountered absolutely continuous probability distribution. More complex experiments, such as those involving
978:
11294:
10994:
10269:
10213:
10111:
9873:
9511:
8348:
5823:
6228:
6089:
11898:
11046:
10555:
10289:
10142:
9817:
9561:
8937:
8781:
5026:
All other possible outcomes then have probability 0. Its cumulative distribution function jumps immediately from 0 to 1.
434:). More commonly, probability distributions are used to compare the relative occurrence of many different random values.
10019:
1833:
Some key concepts and terms, widely used in the literature on the topic of probability distributions, are listed below.
13308:
10787:
10459:
10435:
10014:
9428:
8970:
7934:, a generalization of the Rayleigh distributions for where there is a stationary background signal component. Found in
6315:
6152:
2446:
826:
the probability density function over that interval. An alternative description of the distribution is by means of the
7870:, for the number of each type of categorical outcome, given a fixed number of total outcomes; a generalization of the
6333:. It is not simple to establish that the system has a probability measure, and the main problem is the following. Let
2524:
12681:
12573:
10656:
10533:
10494:
10466:
10440:
10358:
10284:
9707:
9455:
9179:
8893:
8802:
8765:
8666:
8390:
8317:
7700:
7673:
6225:. However, this is not always the case, and there exist phenomena with supports that are actually complicated curves
6049:
5870:
5542:
2924:
1723:
334:
322:
281:
2883:
13286:
12859:
12733:
10863:
10644:
10610:
10476:
10471:
10316:
10124:
9822:
9576:
8294:
8287:
8087:
7703:, which groups by the nature of the outcome being considered (discrete, absolutely continuous, multivariate, etc.)
6311:
3183:
1929:
1795:
827:
3074:
12917:
12578:
12323:
11694:
11284:
10950:
10394:
10307:
10279:
10188:
10137:
10009:
9792:
9757:
7770:
3635:
is commonly used in computer programs that make equal-probability random selections between a number of choices.
3632:
3628:
2808:
2699:
822:
probability of any given value, and the probability that the outcome lies in a given interval can be computed by
382:
212:
148:
3174:{1}, {3}, and {7} are respectively 0.2, 0.5, 0.3. A set not containing any of these points has probability zero.
2753:
1847:: takes values from a sample space; probabilities describe which values and set of values are taken more likely.
12968:
12180:
11987:
11876:
11834:
10408:
10325:
10162:
10086:
9909:
9787:
9762:
9626:
9621:
9616:
8363:
7812:
7802:
5754:
3616:
1775:
260:
121:
11908:
9391:
7723:(Gaussian distribution), for a single such quantity; the most commonly used absolutely continuous distribution
3436:
2471:: a measure of the "fatness" of the tails of a pmf or pdf. The fourth standardized moment of the distribution.
13211:
12170:
11073:
10724:
10590:
10298:
10147:
10079:
10064:
9957:
9931:
9863:
9702:
9596:
9591:
9533:
9518:
9401:
9252:
2008 Third
International Conference on Electric Utility Deregulation and Restructuring and Power Technologies
8373:
7938:
of radio signals due to multipath propagation and in MR images with noise corruption on non-zero NMR signals.
7764:
6600:
6594:
6336:
5761:
31:
7696:
are often inadequate for describing a quantity, while probability distributions are often more appropriate.
5118:
1054:
12762:
12711:
12696:
12686:
12555:
12427:
12394:
12220:
12175:
12005:
10857:
10560:
10550:
10241:
10167:
9868:
9727:
9099:
7884:
7839:
7826:
7795:, for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed total number of
7122:
6964:
of an absolutely continuous random variable, an absolutely continuous random variable must be constructed.
5327:
5035:
4086:
2144:
1817:
1736:
905:
815:
10620:
6553:
3531:
1739:, and the probability distribution is by definition the integral of the probability density function. The
1552:
570:
13274:
13106:
12907:
12831:
12132:
11886:
11555:
11019:
10605:
10600:
10545:
10481:
10425:
10246:
10233:
10024:
9969:
9921:
9712:
9641:
9506:
9396:
9070:
Rabinovich, M.I.; Fabrikant, A.L. (1979). "Stochastic self-modulation of waves in nonequilibrium media".
8683:
8112:
7966:
7888:
7822:
5760:
For a more general definition of density functions and the equivalent absolutely continuous measures see
3210:
is the probability distribution of a random variable that can take on only a countable number of values (
3170:
Figure 4: The probability mass function of a discrete probability distribution. The probabilities of the
1787:
1688:
1114:
can take as argument subsets of the sample space itself, as in the coin toss example, where the function
6155:. What is the probability of observing a state on a certain place of the support (i.e., the red subset)?
5072:
4739:
4405:
446:
A probability distribution is a mathematical description of the probabilities of events, subsets of the
12991:
12963:
12958:
12706:
12465:
12371:
12351:
12259:
11970:
11788:
11271:
11143:
10939:
10739:
10515:
10334:
10116:
10069:
9938:
9914:
9894:
9737:
9611:
9491:
8395:
8108:
7831:
6967:
5993:
4940:
3772:
1851:
1802:
776:, only events that include infinitely many outcomes such as intervals have probability greater than 0.
609:
398:
116:
6281:
6191:
6162:
3631:
that is discrete, and which provides information about the population distribution. Additionally, the
2654:
495:, a set of arbitrary non-numerical values, etc. For example, the sample space of a coin flip could be
412:
is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of
12723:
12491:
12212:
12137:
12066:
11995:
11915:
11903:
11773:
11761:
11754:
11462:
11183:
10851:
10744:
10528:
10489:
10363:
10200:
10044:
9989:
9887:
9851:
9722:
9687:
8736:
3381:
3017:
2958:
2948:
2338:
of the possible values, using their probabilities as their weights; or the continuous analog thereof.
2082:
1767:
1727:
1274:
The above probability function only characterizes a probability distribution if it satisfies all the
823:
527:
232:
9343:
den Dekker, A. J.; Sijbers, J. (2014). "Data distributions in magnetic resonance images: A review".
6727:
4707:{\displaystyle P\left(\bigcup _{i}\Omega _{i}\right)=\sum _{i}P(\Omega _{i})=\sum _{i}P(X=u_{i})=1.}
1614:
1019:
13206:
12973:
12836:
12521:
12486:
12450:
12235:
11586:
11545:
11457:
11148:
10987:
10430:
10218:
9984:
9943:
9858:
9812:
9752:
9717:
9606:
9501:
8740:
8528:
8066:
8062:
8040:
7992:
7974:
7948:
7906:
7880:
7867:
7858:
7796:
7747:
5546:
5373:
3620:
2123:
804:
in the cumulative distribution equals the area under the probability density curve up to the point
291:
286:
175:
8126:). Therefore, the probability distribution function of the position of a particle is described by
1177:
1155:
911:
830:, which describes the probability that the random variable is no larger than a given value (i.e.,
453:
13115:
12728:
12668:
12605:
12243:
12227:
11965:
11827:
11817:
11667:
11581:
10729:
10671:
10342:
10129:
10039:
9994:
9979:
9797:
9747:
9742:
9543:
9523:
8744:
7733:
7681:
6788:{\displaystyle X={\begin{cases}1,&{\text{if }}U<p\\0,&{\text{if }}U\geq p\end{cases}}}
6680:
4986:
3839:
3746:
3386:
3342:
3171:
2385:
2293:
1759:
270:
141:
9899:
8322:
In distribution fitting, therefore, one needs to select a distribution that suits the data well.
1855:: set of possible values (outcomes) of a random variable that occurs with a certain probability.
1638:
13153:
13083:
12876:
12813:
12568:
12455:
11452:
11349:
11256:
11135:
11034:
10595:
10583:
10572:
10454:
10350:
10157:
9601:
9581:
9486:
8310:
8269:
8058:
8044:
8023:
7862:
7808:
7786:
3612:
3604:
3487:
2432:
2112:
2094:
1870:
1513:
1199:
374:
165:
7636:
6677:
has a uniform distribution between 0 and 1. To construct a random
Bernoulli variable for some
2090:): function that gives the probability that a discrete random variable is equal to some value.
13178:
13120:
13063:
12889:
12782:
12691:
12417:
12301:
12160:
12152:
12042:
12034:
11849:
11745:
11723:
11682:
11647:
11614:
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11535:
11490:
11429:
11389:
11191:
11014:
10719:
10676:
10520:
10195:
10049:
10029:
9926:
9496:
8353:
8079:
8027:
7925:
7871:
7792:
7689:
6521:
5746:
5561:
Absolutely continuous probability distributions as defined above are precisely those with an
5066:
3799:
3752:
3608:
2174:
1783:
1546:
The concept of probability function is made more rigorous by defining it as the element of a
484:
306:
265:
170:
136:
7861:, for a single categorical outcome (e.g. yes/no/maybe in a survey); a generalization of the
7492:
2576:
The cumulative distribution function of any real-valued random variable has the properties:
2261:
2235:
2160:(the set of possible values taken by the random variable) can be interpreted as providing a
940:
13101:
12676:
12625:
12601:
12563:
12481:
12460:
12412:
12291:
12269:
12238:
12147:
12024:
11975:
11893:
11866:
11822:
11778:
11540:
11316:
11196:
10769:
10764:
10759:
10754:
10691:
10661:
10540:
10183:
10074:
9677:
9636:
9631:
9528:
9079:
8104:
8075:
8036:
7900:
7708:
4887:
4083:
4079:
3600:
3235:
2920:
2459:: a measure of the extent to which a pmf or pdf "leans" to one side of its mean. The third
2202:
1744:
879:
532:
492:
296:
190:
83:
9974:
8684:"From characteristic function to distribution function: a simple framework for the theory"
2964:
2617:
2584:
794:
8:
13248:
13173:
13096:
12777:
12541:
12534:
12496:
12404:
12384:
12356:
12089:
11955:
11950:
11940:
11932:
11750:
11711:
11601:
11591:
11500:
11279:
11235:
11153:
11078:
10980:
10703:
10228:
10208:
10178:
10152:
10106:
10034:
9846:
9782:
8368:
8052:
7982:
7743:
7737:
7720:
6043:
5988:
5795:
5777:
5750:
5562:
5538:
3624:
2460:
2116:
1859:
1821:
1791:
1752:
1740:
255:
197:
185:
180:
9083:
4355:
1905:
515:(so the sample space can be seen as a numeric set), it is common to distinguish between
13262:
13073:
12927:
12823:
12772:
12648:
12545:
12529:
12506:
12283:
12017:
12000:
11960:
11871:
11766:
11728:
11699:
11659:
11619:
11565:
11482:
11168:
11163:
10911:
10734:
10223:
10004:
9999:
9904:
9841:
9836:
9692:
9682:
9566:
9318:
9273:
9230:
8988:
8866:
8703:
8582:
8340:
8091:
8032:
7912:
7685:
7657:
7527:
7472:
7022:
7002:
6947:
6924:
6660:
6636:
6624:
6606:
6409:
6389:
6222:
6025:
5969:
5961:
5800:
5787:
5718:
5698:
5678:
5568:
5441:
5353:
5333:
5224:
5204:
5184:
5164:
5048:
4966:
4946:
4914:
4785:
4719:
4573:{\displaystyle \Omega _{i}=X^{-1}(u_{i})=\{\omega :X(\omega )=u_{i}\},\,i=0,1,2,\dots }
4385:
4091:
3900:
3880:
3819:
3644:
3599:
Well-known discrete probability distributions used in statistical modeling include the
3322:
3217:
2993:
2504:
2484:
2439:
2365:
2182:
2106:
2048:
2028:
2008:
1988:
1960:
1940:
1664:
1594:
1402:
1254:
1234:
1117:
1097:
362:
242:
131:
71:
48:
10808:
9250:
Chen, P.; Chen, Z.; Bak-Jensen, B. (April 2008). "Probabilistic load flow: A review".
8841:
Khuri, André I. (March 2004). "Applications of Dirac's delta function in statistics".
7894:
7835:
6475:
6429:
5409:
814:
Absolutely continuous probability distributions can be described in several ways. The
13257:
13168:
13138:
13130:
12950:
12941:
12866:
12797:
12653:
12638:
12613:
12501:
12442:
12308:
12296:
11922:
11839:
11783:
11706:
11550:
11472:
11251:
11125:
10869:
10632:
10059:
9802:
9732:
9697:
9646:
9362:
9306:
9296:
9263:
9218:
9208:
9185:
9175:
9151:
9024:
8976:
8966:
8889:
8870:
8858:
8808:
8798:
8761:
8662:
8570:
8560:
8507:
8497:
8471:
8461:
8438:
8428:
8334:
8083:
8070:
8019:
7931:
6330:
5818:
5773:
5565:
cumulative distribution function. In this case, the cumulative distribution function
3627:(a set of observations) is drawn from a larger population, the sample points have an
2349:
1981:
1547:
1275:
480:
301:
207:
106:
9277:
8707:
8557:
A Modern
Introduction to Probability and Statistics : Understanding why and how
7995:
variables; useful e.g. for inferences that involve comparing variances or involving
3202:
Figure 7: ... of a distribution which has both a continuous part and a discrete part
3178:
13193:
13148:
12912:
12899:
12792:
12767:
12701:
12633:
12511:
12119:
12012:
11945:
11858:
11805:
11624:
11495:
11289:
11173:
11088:
11055:
9807:
9481:
9420:
9354:
9255:
9143:
8850:
8695:
8273:
8262:
7960:
7661:
6323:
6310:
One example is shown in the figure to the right, which displays the evolution of a
5865:
2932:
2644:
2335:
1149:
512:
126:
56:
7919:
5757:, i.e. refer to absolutely continuous distributions as continuous distributions.
3190:
2431:: the second moment of the pmf or pdf about the mean; an important measure of the
13110:
12854:
12716:
12643:
12318:
12192:
12165:
12142:
12111:
11738:
11733:
11687:
11417:
11068:
9414:
8854:
8255:
8013:
7970:
7956:
7952:
7903:, for the number of occurrences of a Poisson-type event in a given period of time
5791:
3198:
1843:
1771:
1748:
523:
487:
of a random phenomenon being observed. The sample space may be any set: a set of
202:
153:
27:
Mathematical function for the probability a given outcome occurs in an experiment
12600:
5406:
with coinciding upper and lower limits is always equal to zero. If the interval
13059:
13054:
11517:
11447:
11093:
10901:
10896:
9880:
9358:
7988:
7942:
6631:
6578:
6319:
5783:
3166:
2326:
217:
9310:
9259:
8699:
5534:
is a random variable whose probability distribution is absolutely continuous.
2935:, and thus any cumulative distribution function admits a decomposition as the
2078:: for many random variables with finitely or countably infinitely many values.
13302:
13216:
13183:
13046:
13007:
12818:
12787:
12251:
12205:
11810:
11512:
11339:
11103:
11098:
10503:
10251:
9538:
9222:
8862:
8812:
8574:
8511:
8475:
8442:
8097:
8035:, for a non-negative scaling parameter; conjugate to the rate parameter of a
7935:
7777:
6518:, which might not happen; for example, it could oscillate similar to a sine,
3998:{\displaystyle P(X\in E)=\sum _{\omega \in A}p(\omega )\delta _{\omega }(E),}
3742:
3211:
1812:
1779:
819:
90:
9189:
9147:
8980:
7789:, for the outcome of a single Bernoulli trial (e.g. success/failure, yes/no)
7119:
For example, suppose a random variable that has an exponential distribution
6147:
5537:
There are many examples of absolutely continuous probability distributions:
4931:. This may serve as an alternative definition of discrete random variables.
2115:
where each value has been divided (normalized) by a number of outcomes in a
555:
assigning a probability to each possible outcome (e.g. when throwing a fair
13158:
13091:
13068:
12983:
12313:
11609:
11507:
11442:
11384:
11369:
11306:
11261:
10879:
9366:
8119:
8022:, for a single probability (real number between 0 and 1); conjugate to the
4583:
2443:: the square root of the variance, and hence another measure of dispersion.
2290:: the region where the pmf or pdf is relatively high. Usually has the form
2157:
1763:
488:
447:
390:
317:
227:
111:
7667:
2953:
1041:
13201:
13163:
12846:
12747:
12609:
12422:
12389:
11881:
11798:
11793:
11437:
11394:
11374:
11354:
11344:
11113:
9408:
8843:
International
Journal of Mathematical Education in Science and Technology
8306:
8302:
8007:
7848:
7727:
1934:
1045:
394:
237:
78:
66:
8313:
of occurrence of the magnitude of the phenomenon in a certain interval.
12047:
11527:
11227:
11158:
11108:
11083:
11003:
10891:
8298:
6908:{\displaystyle \Pr(X=1)=\Pr(U<p)=p,\quad \Pr(X=0)=\Pr(U\geq p)=1-p.}
6218:
2936:
386:
366:
95:
41:
4182:{\displaystyle f(x)=\sum _{\omega \in A}p(\omega )\delta (x-\omega ),}
4069:{\displaystyle P_{X}=\sum _{\omega \in A}p(\omega )\delta _{\omega }.}
3816:. Given a discrete probability distribution, there is a countable set
567:, corresponding to the number of dots on the die, has the probability
12200:
12052:
11672:
11467:
11379:
11364:
11359:
11324:
10961:
8965:(Rev. ed.). Cambridge : Cambridge University Press. p. 11.
8460:(Dover ed.). Mineola, N.Y.: Dover Publications. pp. 66â69.
8358:
8123:
8061:, for a vector of probabilities that must sum to 1; conjugate to the
7996:
7751:
7626:{\displaystyle X=F^{\mathit {inv}}(U)={\frac {-1}{\lambda }}\ln(1-U)}
1722:
is applicable to the scenarios where the set of possible outcomes is
431:
9054:
3071:. The pmf allows the computation of probabilities of events such as
11716:
11334:
11211:
11206:
11201:
10956:
10926:
10921:
10916:
10906:
8227:{\textstyle P_{a\leq x\leq b}(t)=\int _{a}^{b}dx\,|\Psi (x,t)|^{2}}
8048:
7895:
Poisson process (events that occur independently with a given rate)
7462:{\displaystyle F^{\mathit {inv}}(u)={\frac {-1}{\lambda }}\ln(1-u)}
5741:
Absolutely continuous distributions ought to be distinguished from
5403:
2467:
2455:
2427:
2358:
1774:
taking on various different values; a multivariate distribution (a
222:
5319:{\displaystyle P\left(a\leq X\leq b\right)=\int _{a}^{b}f(x)\,dx.}
2131:
1718:
Probability distributions usually belong to one of two classes. A
13221:
12922:
8258:
in dimension three. This is a key principle of quantum mechanics.
6159:
Absolutely continuous and discrete distributions with support on
5029:
4875:{\displaystyle X(\omega )=\sum _{i}u_{i}1_{\Omega _{i}}(\omega )}
3312:{\displaystyle P(X\in E)=\sum _{\omega \in A\cap E}P(X=\omega ),}
1685:, that assigns a probability to each of these measurable subsets
3724:{\displaystyle F(x)=P(X\leq x)=\sum _{\omega \leq x}p(\omega ).}
1770:. A univariate distribution gives the probabilities of a single
1503:{\displaystyle P(X\in \bigcup _{i}E_{i})=\sum _{i}P(X\in E_{i})}
1152:, which transform the sample space into a set of numbers (e.g.,
13143:
12124:
12098:
12078:
11329:
11120:
9069:
7920:
Absolute values of vectors with normally distributed components
7693:
2342:
402:
9137:
8268:
Prediction of natural phenomena occurrences based on previous
7767:, for a finite set of values (e.g. the outcome of a fair dice)
4078:
Similarly, discrete distributions can be represented with the
3741:
A discrete probability distribution is often represented with
2164:
that the value of the random variable would equal that sample.
1977:
for a random variable (only for real-valued random variables).
10972:
9415:
Distinguishing probability measure, function and distribution
6329:
This kind of complicated support appears quite frequently in
2156:: function whose value at any given sample (or point) in the
1985:: the inverse of the cumulative distribution function. Gives
511:
To define probability distributions for the specific case of
9012:. New York, USA: Chelsea Publishing Company. pp. 21â24.
8379:
RiemannâStieltjes integral application to probability theory
7943:
Normally distributed quantities operated with sum of squares
5668:{\displaystyle F(x)=P(X\leq x)=\int _{-\infty }^{x}f(t)\,dt}
4448:
be the values it can take with non-zero probability. Denote
2127:: for discrete random variables with a finite set of values.
1392:{\displaystyle P(X\in E)\leq 1\;\forall E\in {\mathcal {A}}}
1332:{\displaystyle P(X\in E)\geq 0\;\forall E\in {\mathcal {A}}}
11063:
8735:
More information and examples can be found in the articles
8496:(2nd ed.). New York: W.H. Freeman and Co. p. 38.
8427:(3rd ed.). Cambridge, UK: Cambridge University Press.
7981:
of normally distributed samples with unknown variance (see
7978:
6781:
3194:
Figure 6: ... of a continuous probability distribution, ...
3011:
904:
The cumulative distribution function is the area under the
784:
556:
8098:
Some specialized applications of probability distributions
7915:, for the time before the next k Poisson-type events occur
7778:
Bernoulli trials (yes/no events, with a given probability)
7110:{\displaystyle {U\leq F(x)}={F^{\mathit {inv}}(U)\leq x}.}
5953:{\displaystyle \{\omega \in \Omega \mid X(\omega )\in A\}}
4963:
has a one-point distribution if it has a possible outcome
2939:
of the three according cumulative distribution functions.
2098:: a table that displays the frequency of various outcomes
1794:. A commonly encountered multivariate distribution is the
7569:
7566:
7405:
7402:
7082:
7079:
6983:
6980:
2140:: for many random variables with uncountably many values.
8602:
Walpole, R.E.; Myers, R.H.; Myers, S.L.; Ye, K. (1999).
7909:, for the time before the next Poisson-type event occurs
6941:. This is a transformation of discrete random variable.
868:{\displaystyle \ {\boldsymbol {\mathcal {P}}}(X<x)\ }
10838:
7714:
7668:
Common probability distributions and their applications
377:
that gives the probabilities of occurrence of possible
9169:
8908:
8494:
Probability and statistics: the science of uncertainty
8132:
8008:
As conjugate prior distributions in
Bayesian inference
7819:
Related to sampling schemes over a finite population:
7728:
Exponential growth (e.g. prices, incomes, populations)
3456:
2919:
Any probability distribution can be decomposed as the
1006:{\displaystyle P\colon {\mathcal {A}}\to \mathbb {R} }
743:
728:
713:
698:
578:
526:. In the discrete case, it is sufficient to specify a
9293:
Statistical methods in hydrology and hydroclimatology
9097:
7639:
7550:
7530:
7495:
7475:
7392:
7181:
7125:
7045:
7025:
7005:
6970:
6950:
6927:
6801:
6715:
6683:
6663:
6639:
6609:
6556:
6524:
6478:
6432:
6412:
6392:
6339:
6284:
6231:
6194:
6165:
6092:
6052:
6028:
5996:
5972:
5913:
5873:
5857:{\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}
5826:
5803:
5721:
5701:
5681:
5591:
5571:
5556:
5464:
5444:
5412:
5376:
5356:
5336:
5247:
5227:
5207:
5187:
5167:
5121:
5075:
5051:
4989:
4969:
4949:
4917:
4890:
4810:
4788:
4742:
4722:
4594:
4456:
4408:
4388:
4358:
4195:
4114:
4094:
4011:
3923:
3903:
3883:
3842:
3822:
3802:
3775:
3755:
3653:
3638:
3534:
3490:
3439:
3389:
3345:
3325:
3244:
3220:
3077:
3020:
2996:
2967:
2886:
2811:
2756:
2702:
2657:
2620:
2587:
2527:
2507:
2487:
2388:
2368:
2296:
2264:
2238:
2205:
2185:
2069:
2051:
2031:
2011:
1991:
1963:
1943:
1908:
1873:
1691:
1667:
1641:
1617:
1597:
1555:
1516:
1426:
1405:
1347:
1287:
1257:
1237:
1231:
the probability that a certain value of the variable
1202:
1180:
1158:
1120:
1100:
1057:
1022:
981:
943:
914:
882:
836:
618:
573:
535:
456:
450:. The sample space, often represented in notation by
12885:
Autoregressive conditional heteroskedasticity (ARCH)
9450:
8888:(3rd ed.). John Wiley & Sons. p. 129.
8330:
6271:{\displaystyle \gamma :\rightarrow \mathbb {R} ^{n}}
6132:{\displaystyle X_{*}\mathbb {P} =\mathbb {P} X^{-1}}
4377:
2942:
9409:
Field Guide to
Continuous Probability Distributions
9207:. Thoman, John W., Jr., 1960-. . pp. 403â406.
9140:
8601:
7955:variables; useful e.g. for inference regarding the
7758:
7656:A frequent problem in statistical simulations (the
2990:specifies the probability distribution for the sum
2475:
12347:
9249:
8226:
7645:
7625:
7536:
7516:
7481:
7461:
7378:
7165:
7109:
7031:
7011:
6991:
6956:
6933:
6907:
6787:
6701:
6669:
6645:
6615:
6568:
6542:
6510:
6464:
6418:
6398:
6378:
6299:
6270:
6209:
6180:
6131:
6078:
6034:
6014:
5978:
5952:
5899:
5856:
5809:
5727:
5707:
5687:
5667:
5577:
5520:
5450:
5430:
5394:
5362:
5342:
5318:
5233:
5213:
5193:
5173:
5153:
5107:
5057:
5018:
4975:
4955:
4923:
4903:
4874:
4794:
4774:
4728:
4706:
4572:
4440:
4394:
4367:
4344:
4181:
4100:
4068:
3997:
3909:
3889:
3869:
3828:
3808:
3788:
3761:
3723:
3588:
3520:
3476:
3425:
3372:
3331:
3311:
3226:
3162:, and all other probabilities in the distribution.
3154:
3063:
3002:
2982:
2908:
2868:
2793:
2742:
2684:
2635:
2602:
2566:
2513:
2493:
2415:
2374:
2314:
2276:
2250:
2218:
2191:
2057:
2037:
2017:
1997:
1969:
1949:
1917:
1894:
1707:
1673:
1653:
1627:
1603:
1583:
1535:
1502:
1411:
1391:
1331:
1263:
1243:
1223:
1188:
1166:
1126:
1106:
1083:
1032:
1005:
958:
929:
894:
867:
762:
600:
547:
471:
13314:Mathematical and quantitative methods (economics)
9342:
9055:Alligood, K.T.; Sauer, T.D.; Yorke, J.A. (1996).
5907:. Given that probabilities of events of the form
969:
13300:
8924:
8641:
7773:, for absolutely continuously distributed values
6872:
6851:
6823:
6802:
5069:probability distribution if there is a function
3214:) which means that the probability of any event
2812:
2758:
2704:
12433:Multivariate adaptive regression splines (MARS)
9202:
8656:
8642:DeGroot, Morris H.; Schervish, Mark J. (2002).
6142:
6079:{\displaystyle ({\mathcal {X}},{\mathcal {A}})}
5900:{\displaystyle ({\mathcal {X}},{\mathcal {A}})}
2132:Absolutely continuous probability distributions
1762:, while a distribution whose sample space is a
1732:absolutely continuous probability distributions
8886:Probability Theory and Mathematical Statistics
8681:
8492:Evans, Michael; Rosenthal, Jeffrey S. (2010).
8491:
8455:
8316:There are many probability distributions (see
7991:, the distribution of the ratio of two scaled
5042:absolutely continuous probability distribution
5030:Absolutely continuous probability distribution
2909:{\displaystyle F:\mathbb {R} \to \mathbb {R} }
2138:Absolutely continuous probability distribution
10988:
10824:
9436:
8529:"1.3.6.1. What is a Probability Distribution"
7977:variable; useful for inference regarding the
6318:) that can be used to model the behaviour of
3736:
416:would take the value 0.5 (1 in 2 or 1/2) for
342:
9205:Physical chemistry for the chemical sciences
8993:: CS1 maint: multiple names: authors list (
6921:has a Bernoulli distribution with parameter
5947:
5914:
5521:{\displaystyle P(X\in A)=\int _{A}f(x)\,dx.}
4536:
4502:
3155:{\displaystyle P(X>9)=1/12+1/18+1/36=1/6}
1530:
1517:
1148:. However, because of the widespread use of
9057:Chaos: an introduction to dynamical systems
9050:
9048:
8911:A First Look at Rigorous Probability Theory
8587:: CS1 maint: numeric names: authors list (
8234:, probability that the particle's position
7692:. For these and many other reasons, simple
6588:
4884:except on a set of probability zero, where
3186:of a discrete probability distribution, ...
2869:{\displaystyle \Pr(a<X\leq b)=F(b)-F(a)}
2743:{\displaystyle \lim _{x\to -\infty }F(x)=0}
11033:
10995:
10981:
10831:
10817:
9443:
9429:
9323:: CS1 maint: location missing publisher (
9235:: CS1 maint: location missing publisher (
9007:
8963:Probability theory : an analytic view
8938:"11. Probability Distributions - Concepts"
8650:
7746:, for a single such quantity whose log is
7736:, for a single such quantity whose log is
7633:. This has an exponential distribution of
4943:. Expressed formally, the random variable
2794:{\displaystyle \lim _{x\to \infty }F(x)=1}
2501:with regard to a probability distribution
1510:for any countable disjoint family of sets
1372:
1312:
349:
335:
11646:
8637:
8635:
8550:
8548:
8188:
7973:variable and the square root of a scaled
6287:
6258:
6197:
6168:
6112:
6104:
6008:
5847:
5658:
5508:
5306:
5147:
5083:
4934:
4542:
4239:
2902:
2894:
1182:
1160:
1077:
1051:as its output, particularly, a number in
999:
9172:Pattern recognition and machine learning
9045:
9010:Foundations of the theory of probability
8604:Probability and statistics for engineers
8082:matrix; conjugate to the inverse of the
7877:Multivariate hypergeometric distribution
6146:
5767:
3477:{\displaystyle p(n)={\tfrac {1}{2^{n}}}}
3197:
3189:
3177:
3165:
2952:
1957:will take a value less than or equal to
1811:
793:
491:, a set of descriptive labels, a set of
385:. It is a mathematical description of a
9117:
8731:
8729:
8554:
8422:
7951:, the distribution of a sum of squared
7524:distribution, then the random variable
6379:{\displaystyle t_{1}\ll t_{2}\ll t_{3}}
1661:whose probability can be measured, and
842:
14:
13301:
12959:KaplanâMeier estimator (product limit)
9373:
8721:
8714:
8632:
8619:
8617:
8615:
8613:
8545:
8425:The Cambridge dictionary of statistics
6623:that are uniformly distributed in the
5458:, the according equality still holds:
5154:{\displaystyle I=\subset \mathbb {R} }
1084:{\displaystyle \subseteq \mathbb {R} }
13032:
12599:
12346:
11645:
11415:
11032:
10976:
10812:
9424:
9290:
9133:
9131:
9129:
9022:
8840:
8792:
8755:
8418:
8416:
7969:, the distribution of the ratio of a
7959:of normally distributed samples (see
7664:that are distributed in a given way.
7166:{\displaystyle F(x)=1-e^{-\lambda x}}
5532:absolutely continuous random variable
4716:It follows that the probability that
3796:be the Dirac measure concentrated at
3014:. For example, the figure shows that
1751:, may demand the use of more general
13269:
12969:Accelerated failure time (AFT) model
10793:
8883:
8825:
8726:
8623:
8523:
8521:
8487:
8485:
8349:Conditional probability distribution
8276:, hail, time in between events, etc.
7715:Linear growth (e.g. errors, offsets)
6569:{\displaystyle t\rightarrow \infty }
5695:is a density of the random variable
3589:{\displaystyle 1/2+1/4+1/8+\dots =1}
3528:, the sum of probabilities would be
1584:{\displaystyle (X,{\mathcal {A}},P)}
1339:, so the probability is non-negative
601:{\displaystyle \ {\tfrac {1}{6}}~).}
13281:
12564:Analysis of variance (ANOVA, anova)
11416:
9374:Vapnik, Vladimir Naumovich (1998).
8610:
7672:For a more comprehensive list, see
3745:, the probability distributions of
1708:{\displaystyle E\in {\mathcal {A}}}
24:
18:Continuous probability distribution
12659:CochranâMantelâHaenszel statistics
11285:Pearson product-moment correlation
9142:, Springer London, pp. 1â11,
9126:
8960:
8935:
8797:. New York: Springer. p. 51.
8760:. New York: Springer. p. 57.
8413:
8354:Empirical probability distribution
8194:
7849:Categorical outcomes (events with
7838:(in some sense, the "opposite" of
7563:
7399:
7076:
7019:, relates to the uniform variable
6977:
6563:
6068:
6058:
5923:
5889:
5879:
5838:
5830:
5636:
5438:is replaced by any measurable set
5108:{\displaystyle f:\mathbb {R} \to }
5099:
4852:
4775:{\displaystyle u_{0},u_{1},\dots }
4648:
4614:
4458:
4441:{\displaystyle u_{0},u_{1},\dots }
2768:
2717:
2070:Discrete probability distributions
1700:
1620:
1567:
1384:
1373:
1324:
1313:
1025:
990:
921:
460:
25:
13325:
9384:
9120:An Introduction to Ergodic Theory
8518:
8482:
8391:List of probability distributions
8318:list of probability distributions
7701:list of probability distributions
7674:List of probability distributions
6992:{\displaystyle F^{\mathit {inv}}}
6015:{\displaystyle X_{*}\mathbb {P} }
4378:Indicator-function representation
3789:{\displaystyle \delta _{\omega }}
3234:can be expressed as a (finite or
3208:discrete probability distribution
2943:Discrete probability distribution
2076:Discrete probability distribution
1766:of dimension 2 or more is called
1720:discrete probability distribution
1611:is the set of possible outcomes,
13280:
13268:
13256:
13243:
13242:
13033:
10878:
10864:cumulative distribution function
10792:
10783:
10782:
9098:Ross, S.M.; Peköz, E.A. (2007).
8782:Lebesgue's decomposition theorem
8555:Dekking, Michel (1946â) (2005).
8333:
8295:Probability distribution fitting
8288:Probability distribution fitting
8286:This section is an excerpt from
8254:in dimension one, and a similar
8088:multivariate normal distribution
7759:Uniformly distributed quantities
6312:system of differential equations
6300:{\displaystyle \mathbb {R} ^{n}}
6210:{\displaystyle \mathbb {N} ^{k}}
6181:{\displaystyle \mathbb {R} ^{k}}
5715:with regard to the distribution
5557:Cumulative distribution function
4782:is zero, and thus one can write
3877:and a probability mass function
3639:Cumulative distribution function
2933:singular continuous distribution
2685:{\displaystyle 0\leq F(x)\leq 1}
2567:{\displaystyle F(x)=P(X\leq x).}
2476:Cumulative distribution function
2168:
1930:Cumulative distribution function
1796:multivariate normal distribution
1681:is the probability function, or
828:cumulative distribution function
55:
12918:Least-squares spectral analysis
10951:probability-generating function
9284:
9243:
9196:
9170:Bishop, Christopher M. (2006).
9163:
9111:
9090:
9063:
9016:
9001:
8954:
8929:
8917:
8909:Jeffrey Seth Rosenthal (2000).
8902:
8877:
8834:
8819:
8786:
8774:
8749:
8675:
7771:Continuous uniform distribution
6850:
6599:Most algorithms are based on a
6151:Figure 8: One solution for the
5962:Kolmogorov's probability axioms
4382:For a discrete random variable
3064:{\displaystyle p(11)=2/36=1/18}
1778:) gives the probabilities of a
441:
11899:Mean-unbiased minimum-variance
11002:
9101:A second course in probability
8925:DeGroot & Schervish (2002)
8595:
8449:
8364:Joint probability distribution
8214:
8209:
8197:
8190:
8161:
8155:
7813:negative binomial distribution
7803:Negative binomial distribution
7750:distributed; the prototypical
7620:
7608:
7581:
7575:
7511:
7499:
7456:
7444:
7417:
7411:
7369:
7357:
7327:
7317:
7305:
7284:
7246:
7208:
7195:
7189:
7135:
7129:
7094:
7088:
7062:
7056:
6887:
6875:
6866:
6854:
6838:
6826:
6817:
6805:
6560:
6537:
6531:
6505:
6479:
6459:
6433:
6316:RabinovichâFabrikant equations
6253:
6250:
6238:
6153:RabinovichâFabrikant equations
6073:
6053:
5938:
5932:
5894:
5874:
5851:
5827:
5655:
5649:
5622:
5610:
5601:
5595:
5505:
5499:
5480:
5468:
5425:
5413:
5303:
5297:
5140:
5128:
5102:
5090:
5087:
5007:
4993:
4869:
4863:
4820:
4814:
4695:
4676:
4657:
4644:
4520:
4514:
4496:
4483:
4339:
4333:
4302:
4290:
4274:
4268:
4236:
4230:
4211:
4199:
4173:
4161:
4155:
4149:
4124:
4118:
4050:
4044:
3989:
3983:
3970:
3964:
3939:
3927:
3858:
3846:
3747:deterministic random variables
3715:
3709:
3684:
3672:
3663:
3657:
3617:negative binomial distribution
3449:
3443:
3420:
3408:
3399:
3393:
3361:
3349:
3303:
3291:
3260:
3248:
3093:
3081:
3030:
3024:
2977:
2971:
2898:
2863:
2857:
2848:
2842:
2833:
2815:
2782:
2776:
2765:
2731:
2725:
2711:
2673:
2667:
2630:
2624:
2597:
2591:
2558:
2546:
2537:
2531:
2404:
2392:
2362:: the q-quantile is the value
1889:
1877:
1836:
1828:
1776:joint probability distribution
1628:{\displaystyle {\mathcal {A}}}
1578:
1556:
1497:
1478:
1459:
1430:
1363:
1351:
1303:
1291:
1218:
1206:
1070:
1058:
1033:{\displaystyle {\mathcal {A}}}
995:
970:General probability definition
859:
847:
691:
675:
666:
650:
641:
625:
592:
122:Collectively exhaustive events
13:
1:
13212:Geographic information system
12428:Simultaneous equations models
8626:A first course in probability
8401:
8374:Quasiprobability distribution
7765:Discrete uniform distribution
6601:pseudorandom number generator
6595:Pseudo-random number sampling
5762:absolutely continuous measure
5395:{\displaystyle a\leq X\leq a}
4911:is the indicator function of
3633:discrete uniform distribution
2199:, it is sometimes denoted as
12395:Coefficient of determination
12006:Uniformly most powerful test
10858:probability density function
9008:Kolmogorov, Andrey (1950) .
8961:W., Stroock, Daniel (1999).
8855:10.1080/00207390310001638313
8406:
7885:sampling without replacement
7840:sampling without replacement
7827:sampling without replacement
6944:For a distribution function
6143:Other kinds of distributions
5966:probability distribution of
5328:probability density function
5326:This is the definition of a
5201:is given by the integral of
5115:such that for each interval
5036:Probability density function
4087:probability density function
2145:Probability density function
2005:such that, with probability
1867:: describes the probability
1818:probability density function
1737:probability density function
1399:, so no probability exceeds
1189:{\displaystyle \mathbb {N} }
1167:{\displaystyle \mathbb {R} }
930:{\displaystyle \ -\infty \ }
906:probability density function
816:probability density function
472:{\displaystyle \ \Omega \ ,}
7:
12964:Proportional hazards models
12908:Spectral density estimation
12890:Vector autoregression (VAR)
12324:Maximum posterior estimator
11556:Randomized controlled trial
9397:Encyclopedia of Mathematics
9376:Statistical Learning Theory
9291:Maity, Rajib (2018-04-30).
8795:Probability and stochastics
8758:Probability and stochastics
8326:
8261:Probabilistic load flow in
8113:natural language processing
8109:statistical language models
7889:hypergeometric distribution
7823:Hypergeometric distribution
7682:kinetic properties of gases
6702:{\displaystyle 0<p<1}
6426:would be equal in interval
5019:{\displaystyle P(X{=}x)=1.}
4736:takes any value except for
3870:{\displaystyle P(X\in A)=1}
3426:{\displaystyle p(x)=P(X=x)}
3373:{\displaystyle P(X\in A)=1}
2416:{\displaystyle P(X<x)=q}
2315:{\displaystyle a<X<b}
1788:hypergeometric distribution
1251:belongs to a certain event
774:continuous random variables
389:phenomenon in terms of its
10:
13330:
12724:Multivariate distributions
11144:Average absolute deviation
10940:moment-generating function
10616:Wrapped asymmetric Laplace
9587:Extended negative binomial
9392:"Probability distribution"
9359:10.1016/j.ejmp.2014.05.002
9335:
8644:Probability and Statistics
8396:List of statistical topics
8285:
8280:
8011:
7887:; a generalization of the
7832:Beta-binomial distribution
7671:
6592:
5771:
5033:
4941:deterministic distribution
3737:Dirac delta representation
2946:
1933:: function evaluating the
1803:moment generating function
1654:{\displaystyle E\subset X}
1635:is the set of all subsets
29:
13309:Probability distributions
13238:
13192:
13129:
13082:
13045:
13041:
13028:
13000:
12982:
12949:
12940:
12898:
12845:
12806:
12755:
12746:
12712:Structural equation model
12667:
12624:
12620:
12595:
12554:
12520:
12474:
12441:
12403:
12370:
12366:
12342:
12282:
12191:
12110:
12074:
12065:
12048:Score/Lagrange multiplier
12033:
11986:
11931:
11857:
11848:
11658:
11654:
11641:
11600:
11574:
11526:
11481:
11463:Sample size determination
11428:
11424:
11411:
11315:
11270:
11244:
11226:
11182:
11134:
11054:
11045:
11041:
11028:
11010:
10935:
10887:
10876:
10852:probability mass function
10847:
10841:probability distributions
10778:
10712:
10670:
10571:
10407:
10385:
10376:
10275:Generalized extreme value
10260:
10095:
10055:Relativistic BreitâWigner
9771:
9668:
9659:
9552:
9472:
9463:
9452:Probability distributions
9260:10.1109/drpt.2008.4523658
8737:Heavy-tailed distribution
8700:10.1017/S0266466600004746
8624:Ross, Sheldon M. (2010).
6999:, an inverse function of
5350:to take any single value
3521:{\displaystyle n=1,2,...}
3382:probability mass function
2959:probability mass function
2949:Probability mass function
2880:Conversely, any function
2083:Probability mass function
1895:{\displaystyle P(X\in E)}
1728:probability mass function
1536:{\displaystyle \{E_{i}\}}
1224:{\displaystyle P(X\in E)}
1094:The probability function
559:, each of the six digits
528:probability mass function
13207:Environmental statistics
12729:Elliptical distributions
12522:Generalized linear model
12451:Simple linear regression
12221:HodgesâLehmann estimator
11678:Probability distribution
11587:Stochastic approximation
11149:Coefficient of variation
9203:Chang, Raymond. (2014).
8826:Cohn, Donald L. (1993).
8741:Long-tailed distribution
8657:Billingsley, P. (1986).
8559:. London, UK: Springer.
8458:Basic probability theory
8384:
8240:will be in the interval
8090:; generalization of the
8069:; generalization of the
8067:multinomial distribution
8063:categorical distribution
8041:exponential distribution
7967:Student's t distribution
7949:Chi-squared distribution
7907:Exponential distribution
7881:multinomial distribution
7868:Multinomial distribution
7859:Categorical distribution
7646:{\displaystyle \lambda }
6589:Random number generation
6386:be instants in time and
5743:continuous distributions
3621:categorical distribution
3339:is a countable set with
2124:Categorical distribution
371:probability distribution
292:Law of total probability
287:Conditional independence
176:Exponential distribution
161:Probability distribution
12867:Cross-correlation (XCF)
12475:Non-standard predictors
11909:LehmannâScheffĂ© theorem
11582:Adaptive clinical trial
10946:characteristic function
10270:Generalized chi-squared
10214:Normal-inverse Gaussian
9148:10.1007/1-84628-168-7_1
9118:Walters, Peter (2000).
9025:"Axioms of Probability"
8745:fat-tailed distribution
8682:Shephard, N.G. (1991).
8659:Probability and measure
8456:Ash, Robert B. (2008).
8423:Everitt, Brian (2006).
8270:frequency distributions
8001:correlation coefficient
7734:Log-normal distribution
7660:) is the generation of
6543:{\displaystyle \sin(t)}
6314:(commonly known as the
3809:{\displaystyle \omega }
3762:{\displaystyle \omega }
1807:characteristic function
271:Conditional probability
13263:Mathematics portal
13084:Engineering statistics
12992:NelsonâAalen estimator
12569:Analysis of covariance
12456:Ordinary least squares
12380:Pearson product-moment
11784:Statistical functional
11695:Empirical distribution
11528:Controlled experiments
11257:Frequency distribution
11035:Descriptive statistics
10582:Univariate (circular)
10143:Generalized hyperbolic
9572:ConwayâMaxwellâPoisson
9562:Beta negative binomial
9378:. John Wiley and Sons.
9254:. pp. 1586â1591.
9174:. New York: Springer.
8793:Erhan, Ăınlar (2011).
8756:Erhan, Ăınlar (2011).
8228:
8059:Dirichlet distribution
8024:Bernoulli distribution
7863:Bernoulli distribution
7809:Geometric distribution
7787:Bernoulli distribution
7647:
7627:
7538:
7518:
7517:{\displaystyle U(0,1)}
7483:
7463:
7380:
7167:
7111:
7033:
7013:
6993:
6958:
6935:
6909:
6789:
6703:
6671:
6647:
6617:
6603:that produces numbers
6570:
6544:
6512:
6466:
6420:
6400:
6380:
6301:
6272:
6211:
6182:
6156:
6133:
6080:
6036:
6016:
5980:
5954:
5901:
5858:
5811:
5747:singular distributions
5729:
5709:
5689:
5669:
5579:
5522:
5452:
5432:
5402:) is zero, because an
5396:
5364:
5344:
5320:
5235:
5215:
5195:
5175:
5155:
5109:
5059:
5020:
4977:
4957:
4935:One-point distribution
4925:
4905:
4876:
4796:
4776:
4730:
4708:
4574:
4442:
4396:
4369:
4346:
4183:
4102:
4070:
3999:
3911:
3891:
3871:
3830:
3810:
3790:
3763:
3725:
3629:empirical distribution
3613:geometric distribution
3605:Bernoulli distribution
3590:
3522:
3478:
3427:
3374:
3333:
3313:
3228:
3203:
3195:
3187:
3175:
3163:
3156:
3065:
3004:
2984:
2910:
2870:
2795:
2744:
2686:
2637:
2604:
2568:
2515:
2495:
2417:
2376:
2316:
2278:
2277:{\displaystyle X<b}
2252:
2251:{\displaystyle X>a}
2220:
2193:
2113:frequency distribution
2095:Frequency distribution
2059:
2039:
2019:
1999:
1971:
1951:
1919:
1896:
1825:
1709:
1675:
1655:
1629:
1605:
1585:
1537:
1504:
1413:
1393:
1333:
1265:
1245:
1225:
1190:
1168:
1128:
1108:
1085:
1034:
1007:
966:as shown in figure 1.
960:
959:{\displaystyle \ x\ ,}
931:
896:
869:
811:
791:underlying equipment.
764:
608:The probability of an
602:
549:
473:
405:of the sample space).
213:Continuous or discrete
166:Bernoulli distribution
13179:Population statistics
13121:System identification
12855:Autocorrelation (ACF)
12783:Exponential smoothing
12697:Discriminant analysis
12692:Canonical correlation
12556:Partition of variance
12418:Regression validation
12262:(JonckheereâTerpstra)
12161:Likelihood-ratio test
11850:Frequentist inference
11762:Locationâscale family
11683:Sampling distribution
11648:Statistical inference
11615:Cross-sectional study
11602:Observational studies
11561:Randomized experiment
11390:Stem-and-leaf display
11192:Central limit theorem
10627:Bivariate (spherical)
10125:Kaniadakis Îș-Gaussian
9417:, Math Stack Exchange
9023:Joyce, David (2014).
8229:
8105:cache language models
8080:non-negative definite
8028:binomial distribution
7926:Rayleigh distribution
7872:binomial distribution
7793:Binomial distribution
7783:Basic distributions:
7690:fundamental particles
7662:pseudo-random numbers
7648:
7628:
7539:
7519:
7484:
7464:
7381:
7173:must be constructed.
7168:
7112:
7034:
7014:
6994:
6959:
6936:
6917:This random variable
6910:
6790:
6704:
6672:
6657:For example, suppose
6648:
6618:
6571:
6545:
6513:
6467:
6421:
6401:
6381:
6302:
6273:
6212:
6183:
6150:
6134:
6081:
6037:
6017:
5981:
5955:
5902:
5859:
5812:
5768:Kolmogorov definition
5755:absolutely continuous
5730:
5710:
5690:
5670:
5580:
5563:absolutely continuous
5523:
5453:
5433:
5397:
5365:
5345:
5321:
5236:
5216:
5196:
5176:
5156:
5110:
5067:absolutely continuous
5060:
5021:
4978:
4958:
4926:
4906:
4904:{\displaystyle 1_{A}}
4877:
4797:
4777:
4731:
4709:
4575:
4443:
4397:
4370:
4347:
4184:
4103:
4071:
4000:
3912:
3892:
3872:
3831:
3811:
3791:
3764:
3726:
3609:binomial distribution
3591:
3523:
3479:
3428:
3375:
3334:
3314:
3229:
3201:
3193:
3181:
3169:
3157:
3066:
3005:
2985:
2956:
2929:absolutely continuous
2911:
2871:
2796:
2745:
2687:
2638:
2605:
2569:
2516:
2496:
2418:
2377:
2317:
2279:
2253:
2221:
2219:{\displaystyle R_{X}}
2194:
2060:
2040:
2020:
2000:
1972:
1952:
1920:
1897:
1815:
1784:binomial distribution
1730:. On the other hand,
1710:
1676:
1656:
1630:
1606:
1586:
1538:
1505:
1414:
1394:
1334:
1266:
1246:
1226:
1191:
1169:
1129:
1109:
1086:
1035:
1008:
961:
932:
897:
895:{\displaystyle \ x\ }
870:
797:
765:
603:
550:
548:{\displaystyle \ p\ }
521:absolutely continuous
474:
171:Binomial distribution
13102:Probabilistic design
12687:Principal components
12530:Exponential families
12482:Nonlinear regression
12461:General linear model
12423:Mixed effects models
12413:Errors and residuals
12390:Confounding variable
12292:Bayesian probability
12270:Van der Waerden test
12260:Ordered alternative
12025:Multiple comparisons
11904:RaoâBlackwellization
11867:Estimating equations
11823:Statistical distance
11541:Factorial experiment
11074:Arithmetic-Geometric
10692:Dirac delta function
10639:Bivariate (toroidal)
10596:Univariate von Mises
10467:Multivariate Laplace
10359:Shifted log-logistic
9708:Continuous Bernoulli
8884:Fisz, Marek (1963).
8720:Chapters 1 and 2 of
8130:
8076:Wishart distribution
8037:Poisson distribution
7901:Poisson distribution
7709:mixture distribution
7637:
7548:
7528:
7493:
7473:
7390:
7179:
7123:
7043:
7023:
7003:
6968:
6948:
6925:
6799:
6713:
6681:
6661:
6637:
6607:
6554:
6522:
6476:
6430:
6410:
6390:
6337:
6282:
6229:
6192:
6163:
6090:
6050:
6026:
5994:
5970:
5911:
5871:
5824:
5801:
5739:Note on terminology:
5719:
5699:
5679:
5589:
5569:
5462:
5442:
5410:
5374:
5354:
5334:
5245:
5225:
5205:
5185:
5165:
5119:
5073:
5049:
4987:
4967:
4947:
4915:
4888:
4808:
4786:
4740:
4720:
4592:
4586:, and for such sets
4454:
4406:
4386:
4356:
4193:
4112:
4092:
4080:Dirac delta function
4009:
3921:
3901:
3881:
3840:
3820:
3800:
3773:
3753:
3651:
3645:jump discontinuities
3601:Poisson distribution
3532:
3488:
3437:
3387:
3343:
3323:
3242:
3218:
3075:
3018:
2994:
2983:{\displaystyle p(S)}
2965:
2884:
2809:
2754:
2700:
2655:
2636:{\displaystyle F(x)}
2618:
2603:{\displaystyle F(x)}
2585:
2525:
2505:
2485:
2463:of the distribution.
2435:of the distribution.
2386:
2366:
2294:
2262:
2236:
2203:
2183:
2049:
2029:
2009:
1989:
1961:
1941:
1906:
1871:
1860:Probability function
1753:probability measures
1745:stochastic processes
1689:
1665:
1639:
1615:
1595:
1553:
1514:
1424:
1403:
1345:
1285:
1255:
1235:
1200:
1178:
1156:
1134:was defined so that
1118:
1098:
1055:
1020:
979:
941:
912:
880:
834:
616:
571:
533:
454:
373:is the mathematical
297:Law of large numbers
266:Marginal probability
191:Poisson distribution
40:Part of a series on
30:For other uses, see
13174:Official statistics
13097:Methods engineering
12778:Seasonal adjustment
12546:Poisson regressions
12466:Bayesian regression
12405:Regression analysis
12385:Partial correlation
12357:Regression analysis
11956:Prediction interval
11951:Likelihood interval
11941:Confidence interval
11933:Interval estimation
11894:Unbiased estimators
11712:Model specification
11592:Up-and-down designs
11280:Partial correlation
11236:Index of dispersion
11154:Interquartile range
10740:Natural exponential
10645:Bivariate von Mises
10611:Wrapped exponential
10477:Multivariate stable
10472:Multivariate normal
9793:Benktander 2nd kind
9788:Benktander 1st kind
9577:Discrete phase-type
9084:1979JETP...50..311R
9072:J. Exp. Theor. Phys
8913:. World Scientific.
8369:Probability measure
8181:
8122:at that point (see
8053:normal distribution
7744:Pareto distribution
7721:Normal distribution
6550:, whose limit when
6044:probability measure
5796:measurable function
5778:Probability measure
5751:Cantor distribution
5645:
5293:
5161:the probability of
3917:is any event, then
3010:of counts from two
2461:standardized moment
2284:or a union thereof.
2162:relative likelihood
2154:probability density
2119:(i.e. sample size).
1865:probability measure
1822:normal distribution
1792:normal distribution
1741:normal distribution
1683:probability measure
256:Complementary event
198:Probability measure
186:Pareto distribution
181:Normal distribution
13194:Spatial statistics
13074:Medical statistics
12974:First hitting time
12928:Whittle likelihood
12579:Degrees of freedom
12574:Multivariate ANOVA
12507:Heteroscedasticity
12319:Bayesian estimator
12284:Bayesian inference
12133:KolmogorovâSmirnov
12018:Randomization test
11988:Testing hypotheses
11961:Tolerance interval
11872:Maximum likelihood
11767:Exponential family
11700:Density estimation
11660:Statistical theory
11620:Natural experiment
11566:Scientific control
11483:Survey methodology
11169:Standard deviation
10912:standard deviation
10395:Rectified Gaussian
10280:Generalized Pareto
10138:Generalized normal
10010:Matrix-exponential
9411:, Gavin E. Crooks.
8688:Econometric Theory
8341:Mathematics portal
8224:
8167:
8092:gamma distribution
8078:, for a symmetric
8033:Gamma distribution
7913:Gamma distribution
7853:possible outcomes)
7686:quantum mechanical
7658:Monte Carlo method
7643:
7623:
7534:
7514:
7479:
7459:
7376:
7374:
7163:
7107:
7029:
7009:
6989:
6954:
6931:
6905:
6785:
6780:
6699:
6667:
6643:
6625:half-open interval
6613:
6566:
6540:
6508:
6462:
6416:
6396:
6376:
6297:
6278:within some space
6268:
6207:
6178:
6157:
6129:
6076:
6032:
6012:
5976:
5950:
5897:
5854:
5807:
5788:probability theory
5725:
5705:
5685:
5665:
5628:
5575:
5518:
5448:
5428:
5392:
5360:
5340:
5316:
5279:
5231:
5211:
5191:
5171:
5151:
5105:
5055:
5016:
4973:
4953:
4921:
4901:
4872:
4835:
4792:
4772:
4726:
4704:
4672:
4640:
4612:
4570:
4438:
4392:
4368:{\displaystyle E.}
4365:
4342:
4329:
4264:
4179:
4145:
4098:
4066:
4040:
3995:
3960:
3907:
3887:
3867:
3826:
3806:
3786:
3759:
3749:. For any outcome
3721:
3705:
3586:
3518:
3474:
3472:
3423:
3370:
3329:
3309:
3287:
3236:countably infinite
3224:
3204:
3196:
3188:
3176:
3164:
3152:
3061:
3000:
2980:
2906:
2866:
2791:
2772:
2740:
2721:
2682:
2633:
2610:is non-decreasing;
2600:
2564:
2511:
2491:
2440:Standard deviation
2413:
2372:
2312:
2274:
2248:
2216:
2189:
2107:Relative frequency
2055:
2035:
2015:
1995:
1967:
1947:
1918:{\displaystyle E,}
1915:
1892:
1826:
1705:
1671:
1651:
1625:
1601:
1581:
1533:
1500:
1474:
1448:
1409:
1389:
1329:
1261:
1241:
1221:
1186:
1164:
1124:
1104:
1081:
1030:
1003:
956:
927:
892:
865:
812:
760:
752:
737:
722:
707:
598:
587:
545:
469:
363:probability theory
307:Boole's inequality
243:Stochastic process
132:Mutual exclusivity
49:Probability theory
13296:
13295:
13234:
13233:
13230:
13229:
13169:National accounts
13139:Actuarial science
13131:Social statistics
13024:
13023:
13020:
13019:
13016:
13015:
12951:Survival function
12936:
12935:
12798:Granger causality
12639:Contingency table
12614:Survival analysis
12591:
12590:
12587:
12586:
12443:Linear regression
12338:
12337:
12334:
12333:
12309:Credible interval
12278:
12277:
12061:
12060:
11877:Method of moments
11746:Parametric family
11707:Statistical model
11637:
11636:
11633:
11632:
11551:Random assignment
11473:Statistical power
11407:
11406:
11403:
11402:
11252:Contingency table
11222:
11221:
11089:Generalized/power
10970:
10969:
10870:quantile function
10806:
10805:
10403:
10402:
10372:
10371:
10263:whose type varies
10209:Normal (Gaussian)
10163:Hyperbolic secant
10112:Exponential power
10015:MaxwellâBoltzmann
9763:Wigner semicircle
9655:
9654:
9627:Parabolic fractal
9617:Negative binomial
9302:978-981-10-8779-0
9269:978-7-900714-13-8
9214:978-1-68015-835-9
9157:978-1-85233-896-1
8646:. Addison-Wesley.
8566:978-1-85233-896-1
8503:978-1-4292-2462-8
8467:978-0-486-46628-6
8434:978-0-511-24688-3
8274:tropical cyclones
8084:covariance matrix
8071:beta distribution
8020:Beta distribution
7932:Rice distribution
7879:, similar to the
7600:
7537:{\displaystyle X}
7482:{\displaystyle U}
7436:
7349:
7032:{\displaystyle U}
7012:{\displaystyle F}
6957:{\displaystyle F}
6934:{\displaystyle p}
6767:
6741:
6670:{\displaystyle U}
6646:{\displaystyle X}
6616:{\displaystyle X}
6419:{\displaystyle O}
6399:{\displaystyle O}
6331:dynamical systems
6035:{\displaystyle X}
5979:{\displaystyle X}
5819:probability space
5810:{\displaystyle X}
5786:formalization of
5784:measure-theoretic
5774:Probability space
5728:{\displaystyle P}
5708:{\displaystyle X}
5688:{\displaystyle f}
5578:{\displaystyle F}
5451:{\displaystyle A}
5363:{\displaystyle a}
5343:{\displaystyle X}
5234:{\displaystyle I}
5214:{\displaystyle f}
5194:{\displaystyle I}
5174:{\displaystyle X}
5058:{\displaystyle X}
4976:{\displaystyle x}
4956:{\displaystyle X}
4924:{\displaystyle A}
4826:
4795:{\displaystyle X}
4729:{\displaystyle X}
4663:
4631:
4603:
4395:{\displaystyle X}
4308:
4249:
4130:
4101:{\displaystyle f}
4025:
3945:
3910:{\displaystyle E}
3890:{\displaystyle p}
3829:{\displaystyle A}
3690:
3471:
3332:{\displaystyle A}
3266:
3227:{\displaystyle E}
3003:{\displaystyle S}
2757:
2703:
2514:{\displaystyle p}
2494:{\displaystyle X}
2375:{\displaystyle x}
2192:{\displaystyle X}
2058:{\displaystyle x}
2038:{\displaystyle X}
2018:{\displaystyle q}
1998:{\displaystyle x}
1982:Quantile function
1970:{\displaystyle x}
1950:{\displaystyle X}
1674:{\displaystyle P}
1604:{\displaystyle X}
1548:probability space
1465:
1439:
1412:{\displaystyle 1}
1276:Kolmogorov axioms
1264:{\displaystyle E}
1244:{\displaystyle X}
1127:{\displaystyle P}
1107:{\displaystyle P}
952:
946:
926:
917:
891:
885:
864:
839:
756:
751:
736:
721:
706:
689:
681:
664:
656:
639:
631:
621:
591:
586:
576:
544:
538:
502:"heads", "tails"
465:
459:
408:For instance, if
359:
358:
261:Joint probability
208:Bernoulli process
107:Probability space
16:(Redirected from
13321:
13284:
13283:
13272:
13271:
13261:
13260:
13246:
13245:
13149:Crime statistics
13043:
13042:
13030:
13029:
12947:
12946:
12913:Fourier analysis
12900:Frequency domain
12880:
12827:
12793:Structural break
12753:
12752:
12702:Cluster analysis
12649:Log-linear model
12622:
12621:
12597:
12596:
12538:
12512:Homoscedasticity
12368:
12367:
12344:
12343:
12263:
12255:
12247:
12246:(KruskalâWallis)
12231:
12216:
12171:Cross validation
12156:
12138:AndersonâDarling
12085:
12072:
12071:
12043:Likelihood-ratio
12035:Parametric tests
12013:Permutation test
11996:1- & 2-tails
11887:Minimum distance
11859:Point estimation
11855:
11854:
11806:Optimal decision
11757:
11656:
11655:
11643:
11642:
11625:Quasi-experiment
11575:Adaptive designs
11426:
11425:
11413:
11412:
11290:Rank correlation
11052:
11051:
11043:
11042:
11030:
11029:
10997:
10990:
10983:
10974:
10973:
10882:
10833:
10826:
10819:
10810:
10809:
10796:
10795:
10786:
10785:
10725:Compound Poisson
10700:
10688:
10657:von MisesâFisher
10653:
10641:
10629:
10591:Circular uniform
10587:
10507:
10451:
10422:
10383:
10382:
10285:MarchenkoâPastur
10148:Geometric stable
10065:Truncated normal
9958:Inverse Gaussian
9864:Hyperexponential
9703:Beta rectangular
9671:bounded interval
9666:
9665:
9534:Discrete uniform
9519:Poisson binomial
9470:
9469:
9445:
9438:
9431:
9422:
9421:
9405:
9379:
9370:
9329:
9328:
9322:
9314:
9288:
9282:
9281:
9247:
9241:
9240:
9234:
9226:
9200:
9194:
9193:
9167:
9161:
9160:
9135:
9124:
9123:
9115:
9109:
9108:
9106:
9094:
9088:
9087:
9067:
9061:
9060:
9052:
9043:
9042:
9040:
9038:
9032:Clark University
9029:
9020:
9014:
9013:
9005:
8999:
8998:
8992:
8984:
8958:
8952:
8951:
8949:
8948:
8936:Bourne, Murray.
8933:
8927:
8921:
8915:
8914:
8906:
8900:
8899:
8881:
8875:
8874:
8838:
8832:
8831:
8823:
8817:
8816:
8790:
8784:
8778:
8772:
8771:
8753:
8747:
8733:
8724:
8718:
8712:
8711:
8679:
8673:
8672:
8654:
8648:
8647:
8639:
8630:
8629:
8621:
8608:
8607:
8606:. Prentice Hall.
8599:
8593:
8592:
8586:
8578:
8552:
8543:
8542:
8540:
8539:
8533:www.itl.nist.gov
8525:
8516:
8515:
8489:
8480:
8479:
8453:
8447:
8446:
8420:
8343:
8338:
8337:
8263:power-flow study
8253:
8239:
8233:
8231:
8230:
8225:
8223:
8222:
8217:
8193:
8180:
8175:
8154:
8153:
7983:Student's t-test
7961:chi-squared test
7852:
7652:
7650:
7649:
7644:
7632:
7630:
7629:
7624:
7601:
7596:
7588:
7574:
7573:
7572:
7543:
7541:
7540:
7535:
7523:
7521:
7520:
7515:
7488:
7486:
7485:
7480:
7468:
7466:
7465:
7460:
7437:
7432:
7424:
7410:
7409:
7408:
7385:
7383:
7382:
7377:
7375:
7350:
7345:
7337:
7323:
7280:
7264:
7263:
7242:
7232:
7231:
7172:
7170:
7169:
7164:
7162:
7161:
7116:
7114:
7113:
7108:
7103:
7087:
7086:
7085:
7065:
7038:
7036:
7035:
7030:
7018:
7016:
7015:
7010:
6998:
6996:
6995:
6990:
6988:
6987:
6986:
6963:
6961:
6960:
6955:
6940:
6938:
6937:
6932:
6914:
6912:
6911:
6906:
6794:
6792:
6791:
6786:
6784:
6783:
6768:
6765:
6742:
6739:
6708:
6706:
6705:
6700:
6676:
6674:
6673:
6668:
6652:
6650:
6649:
6644:
6629:
6622:
6620:
6619:
6614:
6575:
6573:
6572:
6567:
6549:
6547:
6546:
6541:
6517:
6515:
6514:
6511:{\displaystyle }
6509:
6504:
6503:
6491:
6490:
6471:
6469:
6468:
6465:{\displaystyle }
6463:
6458:
6457:
6445:
6444:
6425:
6423:
6422:
6417:
6405:
6403:
6402:
6397:
6385:
6383:
6382:
6377:
6375:
6374:
6362:
6361:
6349:
6348:
6306:
6304:
6303:
6298:
6296:
6295:
6290:
6277:
6275:
6274:
6269:
6267:
6266:
6261:
6216:
6214:
6213:
6208:
6206:
6205:
6200:
6187:
6185:
6184:
6179:
6177:
6176:
6171:
6138:
6136:
6135:
6130:
6128:
6127:
6115:
6107:
6102:
6101:
6085:
6083:
6082:
6077:
6072:
6071:
6062:
6061:
6041:
6039:
6038:
6033:
6021:
6019:
6018:
6013:
6011:
6006:
6005:
5985:
5983:
5982:
5977:
5959:
5957:
5956:
5951:
5906:
5904:
5903:
5898:
5893:
5892:
5883:
5882:
5866:measurable space
5863:
5861:
5860:
5855:
5850:
5842:
5841:
5816:
5814:
5813:
5808:
5794:is defined as a
5734:
5732:
5731:
5726:
5714:
5712:
5711:
5706:
5694:
5692:
5691:
5686:
5674:
5672:
5671:
5666:
5644:
5639:
5584:
5582:
5581:
5576:
5527:
5525:
5524:
5519:
5495:
5494:
5457:
5455:
5454:
5449:
5437:
5435:
5434:
5431:{\displaystyle }
5429:
5401:
5399:
5398:
5393:
5369:
5367:
5366:
5361:
5349:
5347:
5346:
5341:
5325:
5323:
5322:
5317:
5292:
5287:
5275:
5271:
5240:
5238:
5237:
5232:
5220:
5218:
5217:
5212:
5200:
5198:
5197:
5192:
5180:
5178:
5177:
5172:
5160:
5158:
5157:
5152:
5150:
5114:
5112:
5111:
5106:
5086:
5064:
5062:
5061:
5056:
5025:
5023:
5022:
5017:
5003:
4982:
4980:
4979:
4974:
4962:
4960:
4959:
4954:
4930:
4928:
4927:
4922:
4910:
4908:
4907:
4902:
4900:
4899:
4881:
4879:
4878:
4873:
4862:
4861:
4860:
4859:
4845:
4844:
4834:
4801:
4799:
4798:
4793:
4781:
4779:
4778:
4773:
4765:
4764:
4752:
4751:
4735:
4733:
4732:
4727:
4713:
4711:
4710:
4705:
4694:
4693:
4671:
4656:
4655:
4639:
4627:
4623:
4622:
4621:
4611:
4579:
4577:
4576:
4571:
4535:
4534:
4495:
4494:
4482:
4481:
4466:
4465:
4447:
4445:
4444:
4439:
4431:
4430:
4418:
4417:
4401:
4399:
4398:
4393:
4374:
4372:
4371:
4366:
4351:
4349:
4348:
4343:
4328:
4286:
4285:
4263:
4226:
4225:
4188:
4186:
4185:
4180:
4144:
4107:
4105:
4104:
4099:
4075:
4073:
4072:
4067:
4062:
4061:
4039:
4021:
4020:
4004:
4002:
4001:
3996:
3982:
3981:
3959:
3916:
3914:
3913:
3908:
3896:
3894:
3893:
3888:
3876:
3874:
3873:
3868:
3835:
3833:
3832:
3827:
3815:
3813:
3812:
3807:
3795:
3793:
3792:
3787:
3785:
3784:
3768:
3766:
3765:
3760:
3730:
3728:
3727:
3722:
3704:
3595:
3593:
3592:
3587:
3570:
3556:
3542:
3527:
3525:
3524:
3519:
3483:
3481:
3480:
3475:
3473:
3470:
3469:
3457:
3432:
3430:
3429:
3424:
3379:
3377:
3376:
3371:
3338:
3336:
3335:
3330:
3318:
3316:
3315:
3310:
3286:
3233:
3231:
3230:
3225:
3161:
3159:
3158:
3153:
3148:
3134:
3120:
3106:
3070:
3068:
3067:
3062:
3057:
3043:
3009:
3007:
3006:
3001:
2989:
2987:
2986:
2981:
2915:
2913:
2912:
2907:
2905:
2897:
2875:
2873:
2872:
2867:
2800:
2798:
2797:
2792:
2771:
2749:
2747:
2746:
2741:
2720:
2691:
2689:
2688:
2683:
2645:right-continuous
2642:
2640:
2639:
2634:
2609:
2607:
2606:
2601:
2573:
2571:
2570:
2565:
2520:
2518:
2517:
2512:
2500:
2498:
2497:
2492:
2422:
2420:
2419:
2414:
2381:
2379:
2378:
2373:
2336:weighted average
2321:
2319:
2318:
2313:
2283:
2281:
2280:
2275:
2257:
2255:
2254:
2249:
2225:
2223:
2222:
2217:
2215:
2214:
2198:
2196:
2195:
2190:
2064:
2062:
2061:
2056:
2045:will not exceed
2044:
2042:
2041:
2036:
2024:
2022:
2021:
2016:
2004:
2002:
2001:
1996:
1976:
1974:
1973:
1968:
1956:
1954:
1953:
1948:
1924:
1922:
1921:
1916:
1901:
1899:
1898:
1893:
1714:
1712:
1711:
1706:
1704:
1703:
1680:
1678:
1677:
1672:
1660:
1658:
1657:
1652:
1634:
1632:
1631:
1626:
1624:
1623:
1610:
1608:
1607:
1602:
1590:
1588:
1587:
1582:
1571:
1570:
1542:
1540:
1539:
1534:
1529:
1528:
1509:
1507:
1506:
1501:
1496:
1495:
1473:
1458:
1457:
1447:
1418:
1416:
1415:
1410:
1398:
1396:
1395:
1390:
1388:
1387:
1338:
1336:
1335:
1330:
1328:
1327:
1270:
1268:
1267:
1262:
1250:
1248:
1247:
1242:
1230:
1228:
1227:
1222:
1195:
1193:
1192:
1187:
1185:
1173:
1171:
1170:
1165:
1163:
1150:random variables
1147:
1140:
1133:
1131:
1130:
1125:
1113:
1111:
1110:
1105:
1090:
1088:
1087:
1082:
1080:
1039:
1037:
1036:
1031:
1029:
1028:
1012:
1010:
1009:
1004:
1002:
994:
993:
965:
963:
962:
957:
950:
944:
936:
934:
933:
928:
924:
915:
903:
901:
899:
898:
893:
889:
883:
874:
872:
871:
866:
862:
846:
845:
837:
809:
803:
769:
767:
766:
761:
754:
753:
744:
738:
729:
723:
714:
708:
699:
690:
687:
682:
679:
665:
662:
657:
654:
640:
637:
632:
629:
619:
607:
605:
604:
599:
589:
588:
579:
574:
566:
562:
554:
552:
551:
546:
542:
536:
524:random variables
513:random variables
508:
506:
505:
501:
483:of all possible
478:
476:
475:
470:
463:
457:
432:the coin is fair
429:
422:
415:
411:
351:
344:
337:
127:Elementary event
59:
37:
36:
21:
13329:
13328:
13324:
13323:
13322:
13320:
13319:
13318:
13299:
13298:
13297:
13292:
13255:
13226:
13188:
13125:
13111:quality control
13078:
13060:Clinical trials
13037:
13012:
12996:
12984:Hazard function
12978:
12932:
12894:
12878:
12841:
12837:BreuschâGodfrey
12825:
12802:
12742:
12717:Factor analysis
12663:
12644:Graphical model
12616:
12583:
12550:
12536:
12516:
12470:
12437:
12399:
12362:
12361:
12330:
12274:
12261:
12253:
12245:
12229:
12214:
12193:Rank statistics
12187:
12166:Model selection
12154:
12112:Goodness of fit
12106:
12083:
12057:
12029:
11982:
11927:
11916:Median unbiased
11844:
11755:
11688:Order statistic
11650:
11629:
11596:
11570:
11522:
11477:
11420:
11418:Data collection
11399:
11311:
11266:
11240:
11218:
11178:
11130:
11047:Continuous data
11037:
11024:
11006:
11001:
10971:
10966:
10931:
10883:
10874:
10843:
10837:
10807:
10802:
10774:
10750:Maximum entropy
10708:
10696:
10684:
10674:
10666:
10649:
10637:
10625:
10580:
10567:
10504:Matrix-valued:
10501:
10447:
10418:
10410:
10399:
10387:
10378:
10368:
10262:
10256:
10173:
10099:
10097:
10091:
10020:MaxwellâJĂŒttner
9869:Hypoexponential
9775:
9773:
9772:supported on a
9767:
9728:Noncentral beta
9688:BaldingâNichols
9670:
9669:supported on a
9661:
9651:
9554:
9548:
9544:ZipfâMandelbrot
9474:
9465:
9459:
9449:
9390:
9387:
9382:
9338:
9333:
9332:
9316:
9315:
9303:
9289:
9285:
9270:
9248:
9244:
9228:
9227:
9215:
9201:
9197:
9182:
9168:
9164:
9158:
9136:
9127:
9116:
9112:
9104:
9096:Section 1.9 of
9095:
9091:
9068:
9064:
9053:
9046:
9036:
9034:
9027:
9021:
9017:
9006:
9002:
8986:
8985:
8973:
8959:
8955:
8946:
8944:
8942:www.intmath.com
8934:
8930:
8923:Chapter 3.2 of
8922:
8918:
8907:
8903:
8896:
8882:
8878:
8839:
8835:
8824:
8820:
8805:
8791:
8787:
8779:
8775:
8768:
8754:
8750:
8734:
8727:
8719:
8715:
8680:
8676:
8669:
8655:
8651:
8640:
8633:
8622:
8611:
8600:
8596:
8580:
8579:
8567:
8553:
8546:
8537:
8535:
8527:
8526:
8519:
8504:
8490:
8483:
8468:
8454:
8450:
8435:
8421:
8414:
8409:
8404:
8387:
8339:
8332:
8329:
8324:
8323:
8291:
8283:
8256:triple integral
8241:
8235:
8218:
8213:
8212:
8189:
8176:
8171:
8137:
8133:
8131:
8128:
8127:
8100:
8016:
8014:Conjugate prior
8010:
7971:standard normal
7957:sample variance
7953:standard normal
7945:
7922:
7897:
7855:
7850:
7836:PĂłlya urn model
7780:
7761:
7730:
7717:
7688:description of
7677:
7670:
7638:
7635:
7634:
7589:
7587:
7562:
7561:
7557:
7549:
7546:
7545:
7529:
7526:
7525:
7494:
7491:
7490:
7474:
7471:
7470:
7425:
7423:
7398:
7397:
7393:
7391:
7388:
7387:
7373:
7372:
7338:
7336:
7321:
7320:
7278:
7277:
7253:
7249:
7240:
7239:
7221:
7217:
7204:
7182:
7180:
7177:
7176:
7151:
7147:
7124:
7121:
7120:
7075:
7074:
7070:
7069:
7046:
7044:
7041:
7040:
7024:
7021:
7020:
7004:
7001:
7000:
6976:
6975:
6971:
6969:
6966:
6965:
6949:
6946:
6945:
6926:
6923:
6922:
6800:
6797:
6796:
6779:
6778:
6764:
6762:
6753:
6752:
6738:
6736:
6723:
6722:
6714:
6711:
6710:
6682:
6679:
6678:
6662:
6659:
6658:
6638:
6635:
6634:
6632:random variates
6627:
6608:
6605:
6604:
6597:
6591:
6555:
6552:
6551:
6523:
6520:
6519:
6499:
6495:
6486:
6482:
6477:
6474:
6473:
6453:
6449:
6440:
6436:
6431:
6428:
6427:
6411:
6408:
6407:
6391:
6388:
6387:
6370:
6366:
6357:
6353:
6344:
6340:
6338:
6335:
6334:
6291:
6286:
6285:
6283:
6280:
6279:
6262:
6257:
6256:
6230:
6227:
6226:
6201:
6196:
6195:
6193:
6190:
6189:
6172:
6167:
6166:
6164:
6161:
6160:
6145:
6120:
6116:
6111:
6103:
6097:
6093:
6091:
6088:
6087:
6067:
6066:
6057:
6056:
6051:
6048:
6047:
6027:
6024:
6023:
6007:
6001:
5997:
5995:
5992:
5991:
5971:
5968:
5967:
5912:
5909:
5908:
5888:
5887:
5878:
5877:
5872:
5869:
5868:
5846:
5837:
5836:
5825:
5822:
5821:
5802:
5799:
5798:
5792:random variable
5780:
5772:Main articles:
5770:
5720:
5717:
5716:
5700:
5697:
5696:
5680:
5677:
5676:
5640:
5632:
5590:
5587:
5586:
5570:
5567:
5566:
5559:
5490:
5486:
5463:
5460:
5459:
5443:
5440:
5439:
5411:
5408:
5407:
5375:
5372:
5371:
5355:
5352:
5351:
5335:
5332:
5331:
5288:
5283:
5255:
5251:
5246:
5243:
5242:
5226:
5223:
5222:
5206:
5203:
5202:
5186:
5183:
5182:
5166:
5163:
5162:
5146:
5120:
5117:
5116:
5082:
5074:
5071:
5070:
5050:
5047:
5046:
5038:
5032:
4999:
4988:
4985:
4984:
4968:
4965:
4964:
4948:
4945:
4944:
4937:
4916:
4913:
4912:
4895:
4891:
4889:
4886:
4885:
4855:
4851:
4850:
4846:
4840:
4836:
4830:
4809:
4806:
4805:
4787:
4784:
4783:
4760:
4756:
4747:
4743:
4741:
4738:
4737:
4721:
4718:
4717:
4689:
4685:
4667:
4651:
4647:
4635:
4617:
4613:
4607:
4602:
4598:
4593:
4590:
4589:
4530:
4526:
4490:
4486:
4474:
4470:
4461:
4457:
4455:
4452:
4451:
4426:
4422:
4413:
4409:
4407:
4404:
4403:
4387:
4384:
4383:
4380:
4357:
4354:
4353:
4312:
4281:
4277:
4253:
4221:
4217:
4194:
4191:
4190:
4134:
4113:
4110:
4109:
4093:
4090:
4089:
4057:
4053:
4029:
4016:
4012:
4010:
4007:
4006:
3977:
3973:
3949:
3922:
3919:
3918:
3902:
3899:
3898:
3882:
3879:
3878:
3841:
3838:
3837:
3821:
3818:
3817:
3801:
3798:
3797:
3780:
3776:
3774:
3771:
3770:
3754:
3751:
3750:
3739:
3694:
3652:
3649:
3648:
3641:
3566:
3552:
3538:
3533:
3530:
3529:
3489:
3486:
3485:
3465:
3461:
3455:
3438:
3435:
3434:
3388:
3385:
3384:
3344:
3341:
3340:
3324:
3321:
3320:
3270:
3243:
3240:
3239:
3219:
3216:
3215:
3144:
3130:
3116:
3102:
3076:
3073:
3072:
3053:
3039:
3019:
3016:
3015:
2995:
2992:
2991:
2966:
2963:
2962:
2951:
2945:
2901:
2893:
2885:
2882:
2881:
2810:
2807:
2806:
2761:
2755:
2752:
2751:
2707:
2701:
2698:
2697:
2656:
2653:
2652:
2619:
2616:
2615:
2586:
2583:
2582:
2526:
2523:
2522:
2506:
2503:
2502:
2486:
2483:
2482:
2478:
2387:
2384:
2383:
2367:
2364:
2363:
2295:
2292:
2291:
2263:
2260:
2259:
2237:
2234:
2233:
2210:
2206:
2204:
2201:
2200:
2184:
2181:
2180:
2171:
2134:
2072:
2050:
2047:
2046:
2030:
2027:
2026:
2010:
2007:
2006:
1990:
1987:
1986:
1962:
1959:
1958:
1942:
1939:
1938:
1907:
1904:
1903:
1902:that the event
1872:
1869:
1868:
1844:Random variable
1839:
1831:
1772:random variable
1749:continuous time
1699:
1698:
1690:
1687:
1686:
1666:
1663:
1662:
1640:
1637:
1636:
1619:
1618:
1616:
1613:
1612:
1596:
1593:
1592:
1566:
1565:
1554:
1551:
1550:
1524:
1520:
1515:
1512:
1511:
1491:
1487:
1469:
1453:
1449:
1443:
1425:
1422:
1421:
1404:
1401:
1400:
1383:
1382:
1346:
1343:
1342:
1323:
1322:
1286:
1283:
1282:
1256:
1253:
1252:
1236:
1233:
1232:
1201:
1198:
1197:
1181:
1179:
1176:
1175:
1159:
1157:
1154:
1153:
1142:
1135:
1119:
1116:
1115:
1099:
1096:
1095:
1076:
1056:
1053:
1052:
1024:
1023:
1021:
1018:
1017:
998:
989:
988:
980:
977:
976:
972:
942:
939:
938:
913:
910:
909:
881:
878:
877:
876:
841:
840:
835:
832:
831:
805:
799:
742:
727:
712:
697:
686:
678:
661:
653:
636:
628:
617:
614:
613:
577:
572:
569:
568:
564:
560:
534:
531:
530:
503:
499:
497:
496:
455:
452:
451:
444:
430:(assuming that
424:
417:
413:
409:
355:
203:Random variable
154:Bernoulli trial
35:
28:
23:
22:
15:
12:
11:
5:
13327:
13317:
13316:
13311:
13294:
13293:
13291:
13290:
13278:
13266:
13252:
13239:
13236:
13235:
13232:
13231:
13228:
13227:
13225:
13224:
13219:
13214:
13209:
13204:
13198:
13196:
13190:
13189:
13187:
13186:
13181:
13176:
13171:
13166:
13161:
13156:
13151:
13146:
13141:
13135:
13133:
13127:
13126:
13124:
13123:
13118:
13113:
13104:
13099:
13094:
13088:
13086:
13080:
13079:
13077:
13076:
13071:
13066:
13057:
13055:Bioinformatics
13051:
13049:
13039:
13038:
13026:
13025:
13022:
13021:
13018:
13017:
13014:
13013:
13011:
13010:
13004:
13002:
12998:
12997:
12995:
12994:
12988:
12986:
12980:
12979:
12977:
12976:
12971:
12966:
12961:
12955:
12953:
12944:
12938:
12937:
12934:
12933:
12931:
12930:
12925:
12920:
12915:
12910:
12904:
12902:
12896:
12895:
12893:
12892:
12887:
12882:
12874:
12869:
12864:
12863:
12862:
12860:partial (PACF)
12851:
12849:
12843:
12842:
12840:
12839:
12834:
12829:
12821:
12816:
12810:
12808:
12807:Specific tests
12804:
12803:
12801:
12800:
12795:
12790:
12785:
12780:
12775:
12770:
12765:
12759:
12757:
12750:
12744:
12743:
12741:
12740:
12739:
12738:
12737:
12736:
12721:
12720:
12719:
12709:
12707:Classification
12704:
12699:
12694:
12689:
12684:
12679:
12673:
12671:
12665:
12664:
12662:
12661:
12656:
12654:McNemar's test
12651:
12646:
12641:
12636:
12630:
12628:
12618:
12617:
12593:
12592:
12589:
12588:
12585:
12584:
12582:
12581:
12576:
12571:
12566:
12560:
12558:
12552:
12551:
12549:
12548:
12532:
12526:
12524:
12518:
12517:
12515:
12514:
12509:
12504:
12499:
12494:
12492:Semiparametric
12489:
12484:
12478:
12476:
12472:
12471:
12469:
12468:
12463:
12458:
12453:
12447:
12445:
12439:
12438:
12436:
12435:
12430:
12425:
12420:
12415:
12409:
12407:
12401:
12400:
12398:
12397:
12392:
12387:
12382:
12376:
12374:
12364:
12363:
12360:
12359:
12354:
12348:
12340:
12339:
12336:
12335:
12332:
12331:
12329:
12328:
12327:
12326:
12316:
12311:
12306:
12305:
12304:
12299:
12288:
12286:
12280:
12279:
12276:
12275:
12273:
12272:
12267:
12266:
12265:
12257:
12249:
12233:
12230:(MannâWhitney)
12225:
12224:
12223:
12210:
12209:
12208:
12197:
12195:
12189:
12188:
12186:
12185:
12184:
12183:
12178:
12173:
12163:
12158:
12155:(ShapiroâWilk)
12150:
12145:
12140:
12135:
12130:
12122:
12116:
12114:
12108:
12107:
12105:
12104:
12096:
12087:
12075:
12069:
12067:Specific tests
12063:
12062:
12059:
12058:
12056:
12055:
12050:
12045:
12039:
12037:
12031:
12030:
12028:
12027:
12022:
12021:
12020:
12010:
12009:
12008:
11998:
11992:
11990:
11984:
11983:
11981:
11980:
11979:
11978:
11973:
11963:
11958:
11953:
11948:
11943:
11937:
11935:
11929:
11928:
11926:
11925:
11920:
11919:
11918:
11913:
11912:
11911:
11906:
11891:
11890:
11889:
11884:
11879:
11874:
11863:
11861:
11852:
11846:
11845:
11843:
11842:
11837:
11832:
11831:
11830:
11820:
11815:
11814:
11813:
11803:
11802:
11801:
11796:
11791:
11781:
11776:
11771:
11770:
11769:
11764:
11759:
11743:
11742:
11741:
11736:
11731:
11721:
11720:
11719:
11714:
11704:
11703:
11702:
11692:
11691:
11690:
11680:
11675:
11670:
11664:
11662:
11652:
11651:
11639:
11638:
11635:
11634:
11631:
11630:
11628:
11627:
11622:
11617:
11612:
11606:
11604:
11598:
11597:
11595:
11594:
11589:
11584:
11578:
11576:
11572:
11571:
11569:
11568:
11563:
11558:
11553:
11548:
11543:
11538:
11532:
11530:
11524:
11523:
11521:
11520:
11518:Standard error
11515:
11510:
11505:
11504:
11503:
11498:
11487:
11485:
11479:
11478:
11476:
11475:
11470:
11465:
11460:
11455:
11450:
11448:Optimal design
11445:
11440:
11434:
11432:
11422:
11421:
11409:
11408:
11405:
11404:
11401:
11400:
11398:
11397:
11392:
11387:
11382:
11377:
11372:
11367:
11362:
11357:
11352:
11347:
11342:
11337:
11332:
11327:
11321:
11319:
11313:
11312:
11310:
11309:
11304:
11303:
11302:
11297:
11287:
11282:
11276:
11274:
11268:
11267:
11265:
11264:
11259:
11254:
11248:
11246:
11245:Summary tables
11242:
11241:
11239:
11238:
11232:
11230:
11224:
11223:
11220:
11219:
11217:
11216:
11215:
11214:
11209:
11204:
11194:
11188:
11186:
11180:
11179:
11177:
11176:
11171:
11166:
11161:
11156:
11151:
11146:
11140:
11138:
11132:
11131:
11129:
11128:
11123:
11118:
11117:
11116:
11111:
11106:
11101:
11096:
11091:
11086:
11081:
11079:Contraharmonic
11076:
11071:
11060:
11058:
11049:
11039:
11038:
11026:
11025:
11023:
11022:
11017:
11011:
11008:
11007:
11000:
10999:
10992:
10985:
10977:
10968:
10967:
10965:
10964:
10959:
10954:
10948:
10943:
10936:
10933:
10932:
10930:
10929:
10924:
10919:
10914:
10909:
10904:
10899:
10897:central moment
10894:
10888:
10885:
10884:
10877:
10875:
10873:
10872:
10867:
10861:
10855:
10848:
10845:
10844:
10836:
10835:
10828:
10821:
10813:
10804:
10803:
10801:
10800:
10790:
10779:
10776:
10775:
10773:
10772:
10767:
10762:
10757:
10752:
10747:
10745:Locationâscale
10742:
10737:
10732:
10727:
10722:
10716:
10714:
10710:
10709:
10707:
10706:
10701:
10694:
10689:
10681:
10679:
10668:
10667:
10665:
10664:
10659:
10654:
10647:
10642:
10635:
10630:
10623:
10618:
10613:
10608:
10606:Wrapped Cauchy
10603:
10601:Wrapped normal
10598:
10593:
10588:
10577:
10575:
10569:
10568:
10566:
10565:
10564:
10563:
10558:
10556:Normal-inverse
10553:
10548:
10538:
10537:
10536:
10526:
10518:
10513:
10508:
10499:
10498:
10497:
10487:
10479:
10474:
10469:
10464:
10463:
10462:
10452:
10445:
10444:
10443:
10438:
10428:
10423:
10415:
10413:
10405:
10404:
10401:
10400:
10398:
10397:
10391:
10389:
10380:
10374:
10373:
10370:
10369:
10367:
10366:
10361:
10356:
10348:
10340:
10332:
10323:
10314:
10305:
10296:
10287:
10282:
10277:
10272:
10266:
10264:
10258:
10257:
10255:
10254:
10249:
10247:Variance-gamma
10244:
10239:
10231:
10226:
10221:
10216:
10211:
10206:
10198:
10193:
10192:
10191:
10181:
10176:
10171:
10165:
10160:
10155:
10150:
10145:
10140:
10135:
10127:
10122:
10114:
10109:
10103:
10101:
10093:
10092:
10090:
10089:
10087:Wilks's lambda
10084:
10083:
10082:
10072:
10067:
10062:
10057:
10052:
10047:
10042:
10037:
10032:
10027:
10025:Mittag-Leffler
10022:
10017:
10012:
10007:
10002:
9997:
9992:
9987:
9982:
9977:
9972:
9967:
9966:
9965:
9955:
9946:
9941:
9936:
9935:
9934:
9924:
9922:gamma/Gompertz
9919:
9918:
9917:
9912:
9902:
9897:
9892:
9891:
9890:
9878:
9877:
9876:
9871:
9866:
9856:
9855:
9854:
9844:
9839:
9834:
9833:
9832:
9831:
9830:
9820:
9810:
9805:
9800:
9795:
9790:
9785:
9779:
9777:
9774:semi-infinite
9769:
9768:
9766:
9765:
9760:
9755:
9750:
9745:
9740:
9735:
9730:
9725:
9720:
9715:
9710:
9705:
9700:
9695:
9690:
9685:
9680:
9674:
9672:
9663:
9657:
9656:
9653:
9652:
9650:
9649:
9644:
9639:
9634:
9629:
9624:
9619:
9614:
9609:
9604:
9599:
9594:
9589:
9584:
9579:
9574:
9569:
9564:
9558:
9556:
9553:with infinite
9550:
9549:
9547:
9546:
9541:
9536:
9531:
9526:
9521:
9516:
9515:
9514:
9507:Hypergeometric
9504:
9499:
9494:
9489:
9484:
9478:
9476:
9467:
9461:
9460:
9448:
9447:
9440:
9433:
9425:
9419:
9418:
9412:
9406:
9386:
9385:External links
9383:
9381:
9380:
9371:
9353:(7): 725â741.
9346:Physica Medica
9339:
9337:
9334:
9331:
9330:
9301:
9283:
9268:
9242:
9213:
9195:
9180:
9162:
9156:
9125:
9110:
9089:
9062:
9044:
9015:
9000:
8972:978-0521663496
8971:
8953:
8928:
8916:
8901:
8894:
8876:
8849:(2): 185â195.
8833:
8828:Measure theory
8818:
8803:
8785:
8773:
8766:
8748:
8725:
8713:
8694:(4): 519â529.
8674:
8667:
8649:
8631:
8609:
8594:
8565:
8544:
8517:
8502:
8481:
8466:
8448:
8433:
8411:
8410:
8408:
8405:
8403:
8400:
8399:
8398:
8393:
8386:
8383:
8382:
8381:
8376:
8371:
8366:
8361:
8356:
8351:
8345:
8344:
8328:
8325:
8292:
8284:
8282:
8279:
8278:
8277:
8266:
8259:
8221:
8216:
8211:
8208:
8205:
8202:
8199:
8196:
8192:
8187:
8184:
8179:
8174:
8170:
8166:
8163:
8160:
8157:
8152:
8149:
8146:
8143:
8140:
8136:
8116:
8099:
8096:
8095:
8094:
8073:
8056:
8030:
8012:Main article:
8009:
8006:
8005:
8004:
7989:F-distribution
7986:
7964:
7944:
7941:
7940:
7939:
7929:
7921:
7918:
7917:
7916:
7910:
7904:
7896:
7893:
7892:
7891:
7874:
7865:
7854:
7847:
7846:
7845:
7844:
7843:
7829:
7817:
7816:
7815:
7806:
7800:
7790:
7779:
7776:
7775:
7774:
7768:
7760:
7757:
7756:
7755:
7741:
7729:
7726:
7725:
7724:
7716:
7713:
7669:
7666:
7642:
7622:
7619:
7616:
7613:
7610:
7607:
7604:
7599:
7595:
7592:
7586:
7583:
7580:
7577:
7571:
7568:
7565:
7560:
7556:
7553:
7544:is defined by
7533:
7513:
7510:
7507:
7504:
7501:
7498:
7478:
7458:
7455:
7452:
7449:
7446:
7443:
7440:
7435:
7431:
7428:
7422:
7419:
7416:
7413:
7407:
7404:
7401:
7396:
7371:
7368:
7365:
7362:
7359:
7356:
7353:
7348:
7344:
7341:
7335:
7332:
7329:
7326:
7324:
7322:
7319:
7316:
7313:
7310:
7307:
7304:
7301:
7298:
7295:
7292:
7289:
7286:
7283:
7281:
7279:
7276:
7273:
7270:
7267:
7262:
7259:
7256:
7252:
7248:
7245:
7243:
7241:
7238:
7235:
7230:
7227:
7224:
7220:
7216:
7213:
7210:
7207:
7205:
7203:
7200:
7197:
7194:
7191:
7188:
7185:
7184:
7160:
7157:
7154:
7150:
7146:
7143:
7140:
7137:
7134:
7131:
7128:
7106:
7102:
7099:
7096:
7093:
7090:
7084:
7081:
7078:
7073:
7068:
7064:
7061:
7058:
7055:
7052:
7049:
7028:
7008:
6985:
6982:
6979:
6974:
6953:
6930:
6904:
6901:
6898:
6895:
6892:
6889:
6886:
6883:
6880:
6877:
6874:
6871:
6868:
6865:
6862:
6859:
6856:
6853:
6849:
6846:
6843:
6840:
6837:
6834:
6831:
6828:
6825:
6822:
6819:
6816:
6813:
6810:
6807:
6804:
6782:
6777:
6774:
6771:
6763:
6761:
6758:
6755:
6754:
6751:
6748:
6745:
6737:
6735:
6732:
6729:
6728:
6726:
6721:
6718:
6698:
6695:
6692:
6689:
6686:
6666:
6642:
6612:
6593:Main article:
6590:
6587:
6579:ergodic theory
6565:
6562:
6559:
6539:
6536:
6533:
6530:
6527:
6507:
6502:
6498:
6494:
6489:
6485:
6481:
6461:
6456:
6452:
6448:
6443:
6439:
6435:
6415:
6395:
6373:
6369:
6365:
6360:
6356:
6352:
6347:
6343:
6320:Langmuir waves
6294:
6289:
6265:
6260:
6255:
6252:
6249:
6246:
6243:
6240:
6237:
6234:
6204:
6199:
6175:
6170:
6144:
6141:
6126:
6123:
6119:
6114:
6110:
6106:
6100:
6096:
6075:
6070:
6065:
6060:
6055:
6031:
6010:
6004:
6000:
5975:
5949:
5946:
5943:
5940:
5937:
5934:
5931:
5928:
5925:
5922:
5919:
5916:
5896:
5891:
5886:
5881:
5876:
5853:
5849:
5845:
5840:
5835:
5832:
5829:
5806:
5769:
5766:
5724:
5704:
5684:
5664:
5661:
5657:
5654:
5651:
5648:
5643:
5638:
5635:
5631:
5627:
5624:
5621:
5618:
5615:
5612:
5609:
5606:
5603:
5600:
5597:
5594:
5574:
5558:
5555:
5517:
5514:
5511:
5507:
5504:
5501:
5498:
5493:
5489:
5485:
5482:
5479:
5476:
5473:
5470:
5467:
5447:
5427:
5424:
5421:
5418:
5415:
5391:
5388:
5385:
5382:
5379:
5359:
5339:
5315:
5312:
5309:
5305:
5302:
5299:
5296:
5291:
5286:
5282:
5278:
5274:
5270:
5267:
5264:
5261:
5258:
5254:
5250:
5230:
5210:
5190:
5170:
5149:
5145:
5142:
5139:
5136:
5133:
5130:
5127:
5124:
5104:
5101:
5098:
5095:
5092:
5089:
5085:
5081:
5078:
5054:
5034:Main article:
5031:
5028:
5015:
5012:
5009:
5006:
5002:
4998:
4995:
4992:
4972:
4952:
4936:
4933:
4920:
4898:
4894:
4871:
4868:
4865:
4858:
4854:
4849:
4843:
4839:
4833:
4829:
4825:
4822:
4819:
4816:
4813:
4791:
4771:
4768:
4763:
4759:
4755:
4750:
4746:
4725:
4703:
4700:
4697:
4692:
4688:
4684:
4681:
4678:
4675:
4670:
4666:
4662:
4659:
4654:
4650:
4646:
4643:
4638:
4634:
4630:
4626:
4620:
4616:
4610:
4606:
4601:
4597:
4569:
4566:
4563:
4560:
4557:
4554:
4551:
4548:
4545:
4541:
4538:
4533:
4529:
4525:
4522:
4519:
4516:
4513:
4510:
4507:
4504:
4501:
4498:
4493:
4489:
4485:
4480:
4477:
4473:
4469:
4464:
4460:
4437:
4434:
4429:
4425:
4421:
4416:
4412:
4391:
4379:
4376:
4364:
4361:
4352:for any event
4341:
4338:
4335:
4332:
4327:
4324:
4321:
4318:
4315:
4311:
4307:
4304:
4301:
4298:
4295:
4292:
4289:
4284:
4280:
4276:
4273:
4270:
4267:
4262:
4259:
4256:
4252:
4248:
4245:
4242:
4238:
4235:
4232:
4229:
4224:
4220:
4216:
4213:
4210:
4207:
4204:
4201:
4198:
4178:
4175:
4172:
4169:
4166:
4163:
4160:
4157:
4154:
4151:
4148:
4143:
4140:
4137:
4133:
4129:
4126:
4123:
4120:
4117:
4097:
4065:
4060:
4056:
4052:
4049:
4046:
4043:
4038:
4035:
4032:
4028:
4024:
4019:
4015:
3994:
3991:
3988:
3985:
3980:
3976:
3972:
3969:
3966:
3963:
3958:
3955:
3952:
3948:
3944:
3941:
3938:
3935:
3932:
3929:
3926:
3906:
3886:
3866:
3863:
3860:
3857:
3854:
3851:
3848:
3845:
3825:
3805:
3783:
3779:
3758:
3743:Dirac measures
3738:
3735:
3720:
3717:
3714:
3711:
3708:
3703:
3700:
3697:
3693:
3689:
3686:
3683:
3680:
3677:
3674:
3671:
3668:
3665:
3662:
3659:
3656:
3640:
3637:
3585:
3582:
3579:
3576:
3573:
3569:
3565:
3562:
3559:
3555:
3551:
3548:
3545:
3541:
3537:
3517:
3514:
3511:
3508:
3505:
3502:
3499:
3496:
3493:
3468:
3464:
3460:
3454:
3451:
3448:
3445:
3442:
3422:
3419:
3416:
3413:
3410:
3407:
3404:
3401:
3398:
3395:
3392:
3369:
3366:
3363:
3360:
3357:
3354:
3351:
3348:
3328:
3308:
3305:
3302:
3299:
3296:
3293:
3290:
3285:
3282:
3279:
3276:
3273:
3269:
3265:
3262:
3259:
3256:
3253:
3250:
3247:
3223:
3182:Figure 5: The
3151:
3147:
3143:
3140:
3137:
3133:
3129:
3126:
3123:
3119:
3115:
3112:
3109:
3105:
3101:
3098:
3095:
3092:
3089:
3086:
3083:
3080:
3060:
3056:
3052:
3049:
3046:
3042:
3038:
3035:
3032:
3029:
3026:
3023:
2999:
2979:
2976:
2973:
2970:
2957:Figure 3: The
2947:Main article:
2944:
2941:
2904:
2900:
2896:
2892:
2889:
2878:
2877:
2865:
2862:
2859:
2856:
2853:
2850:
2847:
2844:
2841:
2838:
2835:
2832:
2829:
2826:
2823:
2820:
2817:
2814:
2804:
2802:
2790:
2787:
2784:
2781:
2778:
2775:
2770:
2767:
2764:
2760:
2739:
2736:
2733:
2730:
2727:
2724:
2719:
2716:
2713:
2710:
2706:
2695:
2693:
2681:
2678:
2675:
2672:
2669:
2666:
2663:
2660:
2650:
2648:
2632:
2629:
2626:
2623:
2613:
2611:
2599:
2596:
2593:
2590:
2580:
2563:
2560:
2557:
2554:
2551:
2548:
2545:
2542:
2539:
2536:
2533:
2530:
2521:is defined as
2510:
2490:
2477:
2474:
2473:
2472:
2464:
2452:
2444:
2436:
2424:
2412:
2409:
2406:
2403:
2400:
2397:
2394:
2391:
2371:
2355:
2347:
2339:
2327:Expected value
2323:
2311:
2308:
2305:
2302:
2299:
2285:
2273:
2270:
2267:
2247:
2244:
2241:
2227:
2213:
2209:
2188:
2170:
2167:
2166:
2165:
2141:
2133:
2130:
2129:
2128:
2120:
2103:
2101:
2091:
2079:
2071:
2068:
2067:
2066:
2054:
2034:
2014:
1994:
1978:
1966:
1946:
1926:
1914:
1911:
1891:
1888:
1885:
1882:
1879:
1876:
1856:
1848:
1838:
1835:
1830:
1827:
1816:Figure 2: The
1702:
1697:
1694:
1670:
1650:
1647:
1644:
1622:
1600:
1580:
1577:
1574:
1569:
1564:
1561:
1558:
1544:
1543:
1532:
1527:
1523:
1519:
1499:
1494:
1490:
1486:
1483:
1480:
1477:
1472:
1468:
1464:
1461:
1456:
1452:
1446:
1442:
1438:
1435:
1432:
1429:
1419:
1408:
1386:
1381:
1378:
1375:
1371:
1368:
1365:
1362:
1359:
1356:
1353:
1350:
1340:
1326:
1321:
1318:
1315:
1311:
1308:
1305:
1302:
1299:
1296:
1293:
1290:
1260:
1240:
1220:
1217:
1214:
1211:
1208:
1205:
1184:
1162:
1123:
1103:
1079:
1075:
1072:
1069:
1066:
1063:
1060:
1044:, and gives a
1027:
1001:
997:
992:
987:
984:
971:
968:
955:
949:
923:
920:
888:
861:
858:
855:
852:
849:
844:
818:describes the
759:
750:
747:
741:
735:
732:
726:
720:
717:
711:
705:
702:
696:
693:
685:
677:
674:
671:
668:
660:
652:
649:
646:
643:
635:
627:
624:
597:
594:
585:
582:
541:
468:
462:
443:
440:
423:, and 0.5 for
357:
356:
354:
353:
346:
339:
331:
328:
327:
326:
325:
320:
312:
311:
310:
309:
304:
302:Bayes' theorem
299:
294:
289:
284:
276:
275:
274:
273:
268:
263:
258:
250:
249:
248:
247:
246:
245:
240:
235:
233:Observed value
230:
225:
220:
218:Expected value
215:
210:
200:
195:
194:
193:
188:
183:
178:
173:
168:
158:
157:
156:
146:
145:
144:
139:
134:
129:
124:
114:
109:
101:
100:
99:
98:
93:
88:
87:
86:
76:
75:
74:
61:
60:
52:
51:
45:
44:
26:
9:
6:
4:
3:
2:
13326:
13315:
13312:
13310:
13307:
13306:
13304:
13289:
13288:
13279:
13277:
13276:
13267:
13265:
13264:
13259:
13253:
13251:
13250:
13241:
13240:
13237:
13223:
13220:
13218:
13217:Geostatistics
13215:
13213:
13210:
13208:
13205:
13203:
13200:
13199:
13197:
13195:
13191:
13185:
13184:Psychometrics
13182:
13180:
13177:
13175:
13172:
13170:
13167:
13165:
13162:
13160:
13157:
13155:
13152:
13150:
13147:
13145:
13142:
13140:
13137:
13136:
13134:
13132:
13128:
13122:
13119:
13117:
13114:
13112:
13108:
13105:
13103:
13100:
13098:
13095:
13093:
13090:
13089:
13087:
13085:
13081:
13075:
13072:
13070:
13067:
13065:
13061:
13058:
13056:
13053:
13052:
13050:
13048:
13047:Biostatistics
13044:
13040:
13036:
13031:
13027:
13009:
13008:Log-rank test
13006:
13005:
13003:
12999:
12993:
12990:
12989:
12987:
12985:
12981:
12975:
12972:
12970:
12967:
12965:
12962:
12960:
12957:
12956:
12954:
12952:
12948:
12945:
12943:
12939:
12929:
12926:
12924:
12921:
12919:
12916:
12914:
12911:
12909:
12906:
12905:
12903:
12901:
12897:
12891:
12888:
12886:
12883:
12881:
12879:(BoxâJenkins)
12875:
12873:
12870:
12868:
12865:
12861:
12858:
12857:
12856:
12853:
12852:
12850:
12848:
12844:
12838:
12835:
12833:
12832:DurbinâWatson
12830:
12828:
12822:
12820:
12817:
12815:
12814:DickeyâFuller
12812:
12811:
12809:
12805:
12799:
12796:
12794:
12791:
12789:
12788:Cointegration
12786:
12784:
12781:
12779:
12776:
12774:
12771:
12769:
12766:
12764:
12763:Decomposition
12761:
12760:
12758:
12754:
12751:
12749:
12745:
12735:
12732:
12731:
12730:
12727:
12726:
12725:
12722:
12718:
12715:
12714:
12713:
12710:
12708:
12705:
12703:
12700:
12698:
12695:
12693:
12690:
12688:
12685:
12683:
12680:
12678:
12675:
12674:
12672:
12670:
12666:
12660:
12657:
12655:
12652:
12650:
12647:
12645:
12642:
12640:
12637:
12635:
12634:Cohen's kappa
12632:
12631:
12629:
12627:
12623:
12619:
12615:
12611:
12607:
12603:
12598:
12594:
12580:
12577:
12575:
12572:
12570:
12567:
12565:
12562:
12561:
12559:
12557:
12553:
12547:
12543:
12539:
12533:
12531:
12528:
12527:
12525:
12523:
12519:
12513:
12510:
12508:
12505:
12503:
12500:
12498:
12495:
12493:
12490:
12488:
12487:Nonparametric
12485:
12483:
12480:
12479:
12477:
12473:
12467:
12464:
12462:
12459:
12457:
12454:
12452:
12449:
12448:
12446:
12444:
12440:
12434:
12431:
12429:
12426:
12424:
12421:
12419:
12416:
12414:
12411:
12410:
12408:
12406:
12402:
12396:
12393:
12391:
12388:
12386:
12383:
12381:
12378:
12377:
12375:
12373:
12369:
12365:
12358:
12355:
12353:
12350:
12349:
12345:
12341:
12325:
12322:
12321:
12320:
12317:
12315:
12312:
12310:
12307:
12303:
12300:
12298:
12295:
12294:
12293:
12290:
12289:
12287:
12285:
12281:
12271:
12268:
12264:
12258:
12256:
12250:
12248:
12242:
12241:
12240:
12237:
12236:Nonparametric
12234:
12232:
12226:
12222:
12219:
12218:
12217:
12211:
12207:
12206:Sample median
12204:
12203:
12202:
12199:
12198:
12196:
12194:
12190:
12182:
12179:
12177:
12174:
12172:
12169:
12168:
12167:
12164:
12162:
12159:
12157:
12151:
12149:
12146:
12144:
12141:
12139:
12136:
12134:
12131:
12129:
12127:
12123:
12121:
12118:
12117:
12115:
12113:
12109:
12103:
12101:
12097:
12095:
12093:
12088:
12086:
12081:
12077:
12076:
12073:
12070:
12068:
12064:
12054:
12051:
12049:
12046:
12044:
12041:
12040:
12038:
12036:
12032:
12026:
12023:
12019:
12016:
12015:
12014:
12011:
12007:
12004:
12003:
12002:
11999:
11997:
11994:
11993:
11991:
11989:
11985:
11977:
11974:
11972:
11969:
11968:
11967:
11964:
11962:
11959:
11957:
11954:
11952:
11949:
11947:
11944:
11942:
11939:
11938:
11936:
11934:
11930:
11924:
11921:
11917:
11914:
11910:
11907:
11905:
11902:
11901:
11900:
11897:
11896:
11895:
11892:
11888:
11885:
11883:
11880:
11878:
11875:
11873:
11870:
11869:
11868:
11865:
11864:
11862:
11860:
11856:
11853:
11851:
11847:
11841:
11838:
11836:
11833:
11829:
11826:
11825:
11824:
11821:
11819:
11816:
11812:
11811:loss function
11809:
11808:
11807:
11804:
11800:
11797:
11795:
11792:
11790:
11787:
11786:
11785:
11782:
11780:
11777:
11775:
11772:
11768:
11765:
11763:
11760:
11758:
11752:
11749:
11748:
11747:
11744:
11740:
11737:
11735:
11732:
11730:
11727:
11726:
11725:
11722:
11718:
11715:
11713:
11710:
11709:
11708:
11705:
11701:
11698:
11697:
11696:
11693:
11689:
11686:
11685:
11684:
11681:
11679:
11676:
11674:
11671:
11669:
11666:
11665:
11663:
11661:
11657:
11653:
11649:
11644:
11640:
11626:
11623:
11621:
11618:
11616:
11613:
11611:
11608:
11607:
11605:
11603:
11599:
11593:
11590:
11588:
11585:
11583:
11580:
11579:
11577:
11573:
11567:
11564:
11562:
11559:
11557:
11554:
11552:
11549:
11547:
11544:
11542:
11539:
11537:
11534:
11533:
11531:
11529:
11525:
11519:
11516:
11514:
11513:Questionnaire
11511:
11509:
11506:
11502:
11499:
11497:
11494:
11493:
11492:
11489:
11488:
11486:
11484:
11480:
11474:
11471:
11469:
11466:
11464:
11461:
11459:
11456:
11454:
11451:
11449:
11446:
11444:
11441:
11439:
11436:
11435:
11433:
11431:
11427:
11423:
11419:
11414:
11410:
11396:
11393:
11391:
11388:
11386:
11383:
11381:
11378:
11376:
11373:
11371:
11368:
11366:
11363:
11361:
11358:
11356:
11353:
11351:
11348:
11346:
11343:
11341:
11340:Control chart
11338:
11336:
11333:
11331:
11328:
11326:
11323:
11322:
11320:
11318:
11314:
11308:
11305:
11301:
11298:
11296:
11293:
11292:
11291:
11288:
11286:
11283:
11281:
11278:
11277:
11275:
11273:
11269:
11263:
11260:
11258:
11255:
11253:
11250:
11249:
11247:
11243:
11237:
11234:
11233:
11231:
11229:
11225:
11213:
11210:
11208:
11205:
11203:
11200:
11199:
11198:
11195:
11193:
11190:
11189:
11187:
11185:
11181:
11175:
11172:
11170:
11167:
11165:
11162:
11160:
11157:
11155:
11152:
11150:
11147:
11145:
11142:
11141:
11139:
11137:
11133:
11127:
11124:
11122:
11119:
11115:
11112:
11110:
11107:
11105:
11102:
11100:
11097:
11095:
11092:
11090:
11087:
11085:
11082:
11080:
11077:
11075:
11072:
11070:
11067:
11066:
11065:
11062:
11061:
11059:
11057:
11053:
11050:
11048:
11044:
11040:
11036:
11031:
11027:
11021:
11018:
11016:
11013:
11012:
11009:
11005:
10998:
10993:
10991:
10986:
10984:
10979:
10978:
10975:
10963:
10960:
10958:
10955:
10952:
10949:
10947:
10944:
10941:
10938:
10937:
10934:
10928:
10925:
10923:
10920:
10918:
10915:
10913:
10910:
10908:
10905:
10903:
10900:
10898:
10895:
10893:
10890:
10889:
10886:
10881:
10871:
10868:
10865:
10862:
10859:
10856:
10853:
10850:
10849:
10846:
10842:
10834:
10829:
10827:
10822:
10820:
10815:
10814:
10811:
10799:
10791:
10789:
10781:
10780:
10777:
10771:
10768:
10766:
10763:
10761:
10758:
10756:
10753:
10751:
10748:
10746:
10743:
10741:
10738:
10736:
10733:
10731:
10728:
10726:
10723:
10721:
10718:
10717:
10715:
10711:
10705:
10702:
10699:
10695:
10693:
10690:
10687:
10683:
10682:
10680:
10678:
10673:
10669:
10663:
10660:
10658:
10655:
10652:
10648:
10646:
10643:
10640:
10636:
10634:
10631:
10628:
10624:
10622:
10619:
10617:
10614:
10612:
10609:
10607:
10604:
10602:
10599:
10597:
10594:
10592:
10589:
10586:
10585:
10579:
10578:
10576:
10574:
10570:
10562:
10559:
10557:
10554:
10552:
10549:
10547:
10544:
10543:
10542:
10539:
10535:
10532:
10531:
10530:
10527:
10525:
10524:
10519:
10517:
10516:Matrix normal
10514:
10512:
10509:
10506:
10505:
10500:
10496:
10493:
10492:
10491:
10488:
10486:
10485:
10482:Multivariate
10480:
10478:
10475:
10473:
10470:
10468:
10465:
10461:
10458:
10457:
10456:
10453:
10450:
10446:
10442:
10439:
10437:
10434:
10433:
10432:
10429:
10427:
10424:
10421:
10417:
10416:
10414:
10412:
10409:Multivariate
10406:
10396:
10393:
10392:
10390:
10384:
10381:
10375:
10365:
10362:
10360:
10357:
10355:
10353:
10349:
10347:
10345:
10341:
10339:
10337:
10333:
10331:
10329:
10324:
10322:
10320:
10315:
10313:
10311:
10306:
10304:
10302:
10297:
10295:
10293:
10288:
10286:
10283:
10281:
10278:
10276:
10273:
10271:
10268:
10267:
10265:
10261:with support
10259:
10253:
10250:
10248:
10245:
10243:
10240:
10238:
10237:
10232:
10230:
10227:
10225:
10222:
10220:
10217:
10215:
10212:
10210:
10207:
10205:
10204:
10199:
10197:
10194:
10190:
10187:
10186:
10185:
10182:
10180:
10177:
10175:
10174:
10166:
10164:
10161:
10159:
10156:
10154:
10151:
10149:
10146:
10144:
10141:
10139:
10136:
10134:
10133:
10128:
10126:
10123:
10121:
10120:
10115:
10113:
10110:
10108:
10105:
10104:
10102:
10098:on the whole
10094:
10088:
10085:
10081:
10078:
10077:
10076:
10073:
10071:
10070:type-2 Gumbel
10068:
10066:
10063:
10061:
10058:
10056:
10053:
10051:
10048:
10046:
10043:
10041:
10038:
10036:
10033:
10031:
10028:
10026:
10023:
10021:
10018:
10016:
10013:
10011:
10008:
10006:
10003:
10001:
9998:
9996:
9993:
9991:
9988:
9986:
9983:
9981:
9978:
9976:
9973:
9971:
9968:
9964:
9961:
9960:
9959:
9956:
9954:
9952:
9947:
9945:
9942:
9940:
9939:Half-logistic
9937:
9933:
9930:
9929:
9928:
9925:
9923:
9920:
9916:
9913:
9911:
9908:
9907:
9906:
9903:
9901:
9898:
9896:
9895:Folded normal
9893:
9889:
9886:
9885:
9884:
9883:
9879:
9875:
9872:
9870:
9867:
9865:
9862:
9861:
9860:
9857:
9853:
9850:
9849:
9848:
9845:
9843:
9840:
9838:
9835:
9829:
9826:
9825:
9824:
9821:
9819:
9816:
9815:
9814:
9811:
9809:
9806:
9804:
9801:
9799:
9796:
9794:
9791:
9789:
9786:
9784:
9781:
9780:
9778:
9770:
9764:
9761:
9759:
9756:
9754:
9751:
9749:
9746:
9744:
9741:
9739:
9738:Raised cosine
9736:
9734:
9731:
9729:
9726:
9724:
9721:
9719:
9716:
9714:
9711:
9709:
9706:
9704:
9701:
9699:
9696:
9694:
9691:
9689:
9686:
9684:
9681:
9679:
9676:
9675:
9673:
9667:
9664:
9658:
9648:
9645:
9643:
9640:
9638:
9635:
9633:
9630:
9628:
9625:
9623:
9620:
9618:
9615:
9613:
9612:Mixed Poisson
9610:
9608:
9605:
9603:
9600:
9598:
9595:
9593:
9590:
9588:
9585:
9583:
9580:
9578:
9575:
9573:
9570:
9568:
9565:
9563:
9560:
9559:
9557:
9551:
9545:
9542:
9540:
9537:
9535:
9532:
9530:
9527:
9525:
9522:
9520:
9517:
9513:
9510:
9509:
9508:
9505:
9503:
9500:
9498:
9495:
9493:
9492:Beta-binomial
9490:
9488:
9485:
9483:
9480:
9479:
9477:
9471:
9468:
9462:
9457:
9453:
9446:
9441:
9439:
9434:
9432:
9427:
9426:
9423:
9416:
9413:
9410:
9407:
9403:
9399:
9398:
9393:
9389:
9388:
9377:
9372:
9368:
9364:
9360:
9356:
9352:
9348:
9347:
9341:
9340:
9326:
9320:
9312:
9308:
9304:
9298:
9295:. Singapore.
9294:
9287:
9279:
9275:
9271:
9265:
9261:
9257:
9253:
9246:
9238:
9232:
9224:
9220:
9216:
9210:
9206:
9199:
9191:
9187:
9183:
9181:0-387-31073-8
9177:
9173:
9166:
9159:
9153:
9149:
9145:
9141:
9134:
9132:
9130:
9121:
9114:
9103:
9102:
9093:
9085:
9081:
9077:
9073:
9066:
9058:
9051:
9049:
9033:
9026:
9019:
9011:
9004:
8996:
8990:
8982:
8978:
8974:
8968:
8964:
8957:
8943:
8939:
8932:
8926:
8920:
8912:
8905:
8897:
8895:0-471-26250-1
8891:
8887:
8880:
8872:
8868:
8864:
8860:
8856:
8852:
8848:
8844:
8837:
8830:. BirkhÀuser.
8829:
8822:
8814:
8810:
8806:
8804:9780387878591
8800:
8796:
8789:
8783:
8777:
8769:
8767:9780387878584
8763:
8759:
8752:
8746:
8742:
8738:
8732:
8730:
8723:
8722:Vapnik (1998)
8717:
8709:
8705:
8701:
8697:
8693:
8689:
8685:
8678:
8670:
8668:9780471804789
8664:
8660:
8653:
8645:
8638:
8636:
8627:
8620:
8618:
8616:
8614:
8605:
8598:
8590:
8584:
8576:
8572:
8568:
8562:
8558:
8551:
8549:
8534:
8530:
8524:
8522:
8513:
8509:
8505:
8499:
8495:
8488:
8486:
8477:
8473:
8469:
8463:
8459:
8452:
8444:
8440:
8436:
8430:
8426:
8419:
8417:
8412:
8397:
8394:
8392:
8389:
8388:
8380:
8377:
8375:
8372:
8370:
8367:
8365:
8362:
8360:
8357:
8355:
8352:
8350:
8347:
8346:
8342:
8336:
8331:
8321:
8319:
8314:
8312:
8308:
8304:
8300:
8296:
8289:
8275:
8271:
8267:
8264:
8260:
8257:
8252:
8248:
8244:
8238:
8219:
8206:
8203:
8200:
8185:
8182:
8177:
8172:
8168:
8164:
8158:
8150:
8147:
8144:
8141:
8138:
8134:
8125:
8121:
8117:
8114:
8110:
8106:
8102:
8101:
8093:
8089:
8085:
8081:
8077:
8074:
8072:
8068:
8064:
8060:
8057:
8054:
8050:
8046:
8042:
8038:
8034:
8031:
8029:
8025:
8021:
8018:
8017:
8015:
8002:
7999:(the squared
7998:
7994:
7990:
7987:
7984:
7980:
7976:
7972:
7968:
7965:
7962:
7958:
7954:
7950:
7947:
7946:
7937:
7936:Rician fading
7933:
7930:
7927:
7924:
7923:
7914:
7911:
7908:
7905:
7902:
7899:
7898:
7890:
7886:
7882:
7878:
7875:
7873:
7869:
7866:
7864:
7860:
7857:
7856:
7841:
7837:
7833:
7830:
7828:
7824:
7821:
7820:
7818:
7814:
7810:
7807:
7804:
7801:
7798:
7794:
7791:
7788:
7785:
7784:
7782:
7781:
7772:
7769:
7766:
7763:
7762:
7753:
7749:
7748:exponentially
7745:
7742:
7739:
7735:
7732:
7731:
7722:
7719:
7718:
7712:
7710:
7704:
7702:
7697:
7695:
7691:
7687:
7683:
7675:
7665:
7663:
7659:
7654:
7640:
7617:
7614:
7611:
7605:
7602:
7597:
7593:
7590:
7584:
7578:
7558:
7554:
7551:
7531:
7508:
7505:
7502:
7496:
7476:
7453:
7450:
7447:
7441:
7438:
7433:
7429:
7426:
7420:
7414:
7394:
7366:
7363:
7360:
7354:
7351:
7346:
7342:
7339:
7333:
7330:
7325:
7314:
7311:
7308:
7302:
7299:
7296:
7293:
7290:
7287:
7282:
7274:
7271:
7268:
7265:
7260:
7257:
7254:
7250:
7244:
7236:
7233:
7228:
7225:
7222:
7218:
7214:
7211:
7206:
7201:
7198:
7192:
7186:
7174:
7158:
7155:
7152:
7148:
7144:
7141:
7138:
7132:
7126:
7117:
7104:
7100:
7097:
7091:
7071:
7066:
7059:
7053:
7050:
7047:
7026:
7006:
6972:
6951:
6942:
6928:
6920:
6915:
6902:
6899:
6896:
6893:
6890:
6884:
6881:
6878:
6869:
6863:
6860:
6857:
6847:
6844:
6841:
6835:
6832:
6829:
6820:
6814:
6811:
6808:
6775:
6772:
6769:
6759:
6756:
6749:
6746:
6743:
6733:
6730:
6724:
6719:
6716:
6696:
6693:
6690:
6687:
6684:
6664:
6655:
6640:
6633:
6626:
6610:
6602:
6596:
6586:
6582:
6580:
6557:
6534:
6528:
6525:
6500:
6496:
6492:
6487:
6483:
6454:
6450:
6446:
6441:
6437:
6413:
6393:
6371:
6367:
6363:
6358:
6354:
6350:
6345:
6341:
6332:
6327:
6325:
6321:
6317:
6313:
6308:
6292:
6263:
6247:
6244:
6241:
6235:
6232:
6224:
6220:
6202:
6173:
6154:
6149:
6140:
6124:
6121:
6117:
6108:
6098:
6094:
6063:
6045:
6042:, which is a
6029:
6002:
5998:
5990:
5989:image measure
5986:
5973:
5963:
5944:
5941:
5935:
5929:
5926:
5920:
5917:
5884:
5867:
5843:
5833:
5820:
5804:
5797:
5793:
5789:
5785:
5779:
5775:
5765:
5763:
5758:
5756:
5752:
5748:
5744:
5740:
5736:
5722:
5702:
5682:
5662:
5659:
5652:
5646:
5641:
5633:
5629:
5625:
5619:
5616:
5613:
5607:
5604:
5598:
5592:
5585:has the form
5572:
5564:
5554:
5552:
5548:
5544:
5540:
5535:
5533:
5528:
5515:
5512:
5509:
5502:
5496:
5491:
5487:
5483:
5477:
5474:
5471:
5465:
5445:
5422:
5419:
5416:
5405:
5389:
5386:
5383:
5380:
5377:
5357:
5337:
5329:
5313:
5310:
5307:
5300:
5294:
5289:
5284:
5280:
5276:
5272:
5268:
5265:
5262:
5259:
5256:
5252:
5248:
5228:
5208:
5188:
5181:belonging to
5168:
5143:
5137:
5134:
5131:
5125:
5122:
5096:
5093:
5079:
5076:
5068:
5052:
5043:
5037:
5027:
5013:
5010:
5004:
5000:
4996:
4990:
4970:
4950:
4942:
4932:
4918:
4896:
4892:
4882:
4866:
4856:
4847:
4841:
4837:
4831:
4827:
4823:
4817:
4811:
4803:
4789:
4769:
4766:
4761:
4757:
4753:
4748:
4744:
4723:
4714:
4701:
4698:
4690:
4686:
4682:
4679:
4673:
4668:
4664:
4660:
4652:
4641:
4636:
4632:
4628:
4624:
4618:
4608:
4604:
4599:
4595:
4587:
4585:
4584:disjoint sets
4580:
4567:
4564:
4561:
4558:
4555:
4552:
4549:
4546:
4543:
4539:
4531:
4527:
4523:
4517:
4511:
4508:
4505:
4499:
4491:
4487:
4478:
4475:
4471:
4467:
4462:
4449:
4435:
4432:
4427:
4423:
4419:
4414:
4410:
4389:
4375:
4362:
4359:
4336:
4330:
4325:
4322:
4319:
4316:
4313:
4309:
4305:
4299:
4296:
4293:
4287:
4282:
4278:
4271:
4265:
4260:
4257:
4254:
4250:
4246:
4243:
4240:
4233:
4227:
4222:
4218:
4214:
4208:
4205:
4202:
4196:
4176:
4170:
4167:
4164:
4158:
4152:
4146:
4141:
4138:
4135:
4131:
4127:
4121:
4115:
4095:
4088:
4085:
4081:
4076:
4063:
4058:
4054:
4047:
4041:
4036:
4033:
4030:
4026:
4022:
4017:
4013:
4005:or in short,
3992:
3986:
3978:
3974:
3967:
3961:
3956:
3953:
3950:
3946:
3942:
3936:
3933:
3930:
3924:
3904:
3884:
3864:
3861:
3855:
3852:
3849:
3843:
3823:
3803:
3781:
3777:
3756:
3748:
3744:
3734:
3731:
3718:
3712:
3706:
3701:
3698:
3695:
3691:
3687:
3681:
3678:
3675:
3669:
3666:
3660:
3654:
3646:
3636:
3634:
3630:
3626:
3622:
3618:
3614:
3610:
3606:
3602:
3597:
3583:
3580:
3577:
3574:
3571:
3567:
3563:
3560:
3557:
3553:
3549:
3546:
3543:
3539:
3535:
3515:
3512:
3509:
3506:
3503:
3500:
3497:
3494:
3491:
3466:
3462:
3458:
3452:
3446:
3440:
3417:
3414:
3411:
3405:
3402:
3396:
3390:
3383:
3367:
3364:
3358:
3355:
3352:
3346:
3326:
3306:
3300:
3297:
3294:
3288:
3283:
3280:
3277:
3274:
3271:
3267:
3263:
3257:
3254:
3251:
3245:
3237:
3221:
3213:
3212:almost surely
3209:
3200:
3192:
3185:
3180:
3173:
3168:
3149:
3145:
3141:
3138:
3135:
3131:
3127:
3124:
3121:
3117:
3113:
3110:
3107:
3103:
3099:
3096:
3090:
3087:
3084:
3078:
3058:
3054:
3050:
3047:
3044:
3040:
3036:
3033:
3027:
3021:
3013:
2997:
2974:
2968:
2960:
2955:
2950:
2940:
2938:
2934:
2930:
2926:
2922:
2917:
2890:
2887:
2860:
2854:
2851:
2845:
2839:
2836:
2830:
2827:
2824:
2821:
2818:
2805:
2803:
2788:
2785:
2779:
2773:
2762:
2737:
2734:
2728:
2722:
2714:
2708:
2696:
2694:
2679:
2676:
2670:
2664:
2661:
2658:
2651:
2649:
2646:
2627:
2621:
2614:
2612:
2594:
2588:
2581:
2579:
2578:
2577:
2574:
2561:
2555:
2552:
2549:
2543:
2540:
2534:
2528:
2508:
2488:
2470:
2469:
2465:
2462:
2458:
2457:
2453:
2450:
2449:
2445:
2442:
2441:
2437:
2434:
2430:
2429:
2425:
2410:
2407:
2401:
2398:
2395:
2389:
2369:
2361:
2360:
2356:
2353:
2352:
2348:
2345:
2344:
2340:
2337:
2333:
2329:
2328:
2324:
2309:
2306:
2303:
2300:
2297:
2289:
2286:
2271:
2268:
2265:
2245:
2242:
2239:
2231:
2228:
2211:
2207:
2186:
2178:
2177:
2173:
2172:
2169:Related terms
2163:
2159:
2155:
2151:
2147:
2146:
2142:
2139:
2136:
2135:
2126:
2125:
2121:
2118:
2114:
2110:
2108:
2104:
2099:
2097:
2096:
2092:
2089:
2085:
2084:
2080:
2077:
2074:
2073:
2052:
2032:
2012:
1992:
1984:
1983:
1979:
1964:
1944:
1936:
1932:
1931:
1927:
1912:
1909:
1886:
1883:
1880:
1874:
1866:
1862:
1861:
1857:
1854:
1853:
1849:
1846:
1845:
1841:
1840:
1834:
1823:
1820:(pdf) of the
1819:
1814:
1810:
1808:
1804:
1799:
1797:
1793:
1789:
1785:
1781:
1780:random vector
1777:
1773:
1769:
1765:
1761:
1756:
1754:
1750:
1746:
1742:
1738:
1733:
1729:
1725:
1721:
1716:
1695:
1692:
1684:
1668:
1648:
1645:
1642:
1598:
1575:
1572:
1562:
1559:
1549:
1525:
1521:
1492:
1488:
1484:
1481:
1475:
1470:
1466:
1462:
1454:
1450:
1444:
1440:
1436:
1433:
1427:
1420:
1406:
1379:
1376:
1369:
1366:
1360:
1357:
1354:
1348:
1341:
1319:
1316:
1309:
1306:
1300:
1297:
1294:
1288:
1281:
1280:
1279:
1277:
1272:
1258:
1238:
1215:
1212:
1209:
1203:
1151:
1146:(tails) = 0.5
1145:
1139:(heads) = 0.5
1138:
1121:
1101:
1092:
1073:
1067:
1064:
1061:
1050:
1047:
1043:
1016:
985:
982:
967:
953:
947:
918:
907:
886:
856:
853:
850:
829:
825:
821:
820:infinitesimal
817:
808:
802:
796:
792:
788:
786:
782:
777:
775:
770:
757:
748:
745:
739:
733:
730:
724:
718:
715:
709:
703:
700:
694:
683:
672:
669:
658:
647:
644:
633:
622:
611:
595:
583:
580:
558:
539:
529:
525:
522:
518:
514:
509:
494:
490:
486:
482:
466:
449:
439:
435:
433:
427:
420:
406:
404:
400:
396:
395:probabilities
392:
388:
384:
380:
376:
372:
368:
364:
352:
347:
345:
340:
338:
333:
332:
330:
329:
324:
321:
319:
316:
315:
314:
313:
308:
305:
303:
300:
298:
295:
293:
290:
288:
285:
283:
280:
279:
278:
277:
272:
269:
267:
264:
262:
259:
257:
254:
253:
252:
251:
244:
241:
239:
236:
234:
231:
229:
226:
224:
221:
219:
216:
214:
211:
209:
206:
205:
204:
201:
199:
196:
192:
189:
187:
184:
182:
179:
177:
174:
172:
169:
167:
164:
163:
162:
159:
155:
152:
151:
150:
147:
143:
140:
138:
135:
133:
130:
128:
125:
123:
120:
119:
118:
115:
113:
110:
108:
105:
104:
103:
102:
97:
94:
92:
91:Indeterminism
89:
85:
82:
81:
80:
77:
73:
70:
69:
68:
65:
64:
63:
62:
58:
54:
53:
50:
47:
46:
43:
39:
38:
33:
19:
13285:
13273:
13254:
13247:
13159:Econometrics
13109: /
13092:Chemometrics
13069:Epidemiology
13062: /
13035:Applications
12877:ARIMA model
12824:Q-statistic
12773:Stationarity
12669:Multivariate
12612: /
12608: /
12606:Multivariate
12604: /
12544: /
12540: /
12314:Bayes factor
12213:Signed rank
12125:
12099:
12091:
12079:
11774:Completeness
11677:
11610:Cohort study
11508:Opinion poll
11443:Missing data
11430:Study design
11385:Scatter plot
11307:Scatter plot
11300:Spearman's Ï
11262:Grouped data
10840:
10697:
10685:
10651:Multivariate
10650:
10638:
10626:
10621:Wrapped LĂ©vy
10581:
10529:Matrix gamma
10522:
10502:
10490:Normal-gamma
10483:
10449:Continuous:
10448:
10419:
10364:Tukey lambda
10351:
10343:
10338:-exponential
10335:
10327:
10318:
10309:
10300:
10294:-exponential
10291:
10235:
10202:
10169:
10131:
10118:
10045:Poly-Weibull
9990:Log-logistic
9950:
9949:Hotelling's
9881:
9723:Logit-normal
9597:GaussâKuzmin
9592:FloryâSchulz
9473:with finite
9451:
9395:
9375:
9350:
9344:
9292:
9286:
9251:
9245:
9204:
9198:
9171:
9165:
9139:
9119:
9113:
9100:
9092:
9075:
9071:
9065:
9056:
9035:. Retrieved
9031:
9018:
9009:
9003:
8962:
8956:
8945:. Retrieved
8941:
8931:
8919:
8910:
8904:
8885:
8879:
8846:
8842:
8836:
8827:
8821:
8794:
8788:
8776:
8757:
8751:
8716:
8691:
8687:
8677:
8658:
8652:
8643:
8625:
8603:
8597:
8556:
8536:. Retrieved
8532:
8493:
8457:
8451:
8424:
8315:
8293:
8250:
8246:
8242:
8236:
8120:wavefunction
7883:, but using
7754:distribution
7705:
7698:
7678:
7655:
7175:
7118:
6943:
6918:
6916:
6709:, we define
6656:
6598:
6583:
6328:
6309:
6158:
5965:
5781:
5759:
5742:
5738:
5737:
5560:
5536:
5531:
5529:
5041:
5039:
4938:
4883:
4804:
4715:
4588:
4581:
4450:
4381:
4189:which means
4077:
3740:
3732:
3642:
3598:
3207:
3205:
2918:
2879:
2575:
2479:
2466:
2454:
2447:
2438:
2426:
2357:
2350:
2341:
2331:
2325:
2287:
2229:
2175:
2161:
2158:sample space
2153:
2149:
2143:
2137:
2122:
2109:distribution
2105:
2093:
2087:
2081:
2075:
1980:
1928:
1864:
1858:
1850:
1842:
1832:
1800:
1768:multivariate
1764:vector space
1757:
1731:
1719:
1717:
1682:
1545:
1273:
1143:
1136:
1093:
1048:
1014:
973:
813:
806:
800:
789:
780:
778:
773:
771:
520:
516:
510:
489:real numbers
448:sample space
445:
442:Introduction
436:
425:
418:
407:
391:sample space
378:
370:
360:
323:Tree diagram
318:Venn diagram
282:Independence
228:Markov chain
160:
112:Sample space
32:Distribution
13287:WikiProject
13202:Cartography
13164:Jurimetrics
13116:Reliability
12847:Time domain
12826:(LjungâBox)
12748:Time-series
12626:Categorical
12610:Time-series
12602:Categorical
12537:(Bernoulli)
12372:Correlation
12352:Correlation
12148:JarqueâBera
12120:Chi-squared
11882:M-estimator
11835:Asymptotics
11779:Sufficiency
11546:Interaction
11458:Replication
11438:Effect size
11395:Violin plot
11375:Radar chart
11355:Forest plot
11345:Correlogram
11295:Kendall's Ï
10735:Exponential
10584:directional
10573:Directional
10460:Generalized
10431:Multinomial
10386:continuous-
10326:Kaniadakis
10317:Kaniadakis
10308:Kaniadakis
10299:Kaniadakis
10290:Kaniadakis
10242:TracyâWidom
10219:Skew normal
10201:Noncentral
9985:Log-Laplace
9963:Generalized
9944:Half-normal
9910:Generalized
9874:Logarithmic
9859:Exponential
9813:Chi-squared
9753:U-quadratic
9718:Kumaraswamy
9660:Continuous
9607:Logarithmic
9502:Categorical
9122:. Springer.
9078:: 617â629.
9059:. Springer.
9037:December 5,
8303:probability
7993:chi squared
7975:chi squared
7799:occurrences
7797:independent
7740:distributed
6086:satisfying
5547:chi-squared
4084:generalized
2100:in a sample
1935:probability
1837:Basic terms
1829:Terminology
1747:defined in
1278:, that is:
1049:probability
1046:real number
1015:input space
824:integrating
498: Ω =
238:Random walk
79:Determinism
67:Probability
13303:Categories
13154:Demography
12872:ARMA model
12677:Regression
12254:(Friedman)
12215:(Wilcoxon)
12153:Normality
12143:Lilliefors
12090:Student's
11966:Resampling
11840:Robustness
11828:divergence
11818:Efficiency
11756:(monotone)
11751:Likelihood
11668:Population
11501:Stratified
11453:Population
11272:Dependence
11228:Count data
11159:Percentile
11136:Dispersion
11069:Arithmetic
11004:Statistics
10892:raw moment
10839:Theory of
10730:Elliptical
10686:Degenerate
10672:Degenerate
10420:Discrete:
10379:univariate
10234:Student's
10189:Asymmetric
10168:Johnson's
10096:supported
10040:Phase-type
9995:Log-normal
9980:Log-Cauchy
9970:Kolmogorov
9888:Noncentral
9818:Noncentral
9798:Beta prime
9748:Triangular
9743:Reciprocal
9713:IrwinâHall
9662:univariate
9642:YuleâSimon
9524:Rademacher
9466:univariate
9311:1038418263
8947:2020-09-10
8628:. Pearson.
8538:2020-09-10
8402:References
8107:and other
6628:[0, 1)
6219:hypercubes
5370:(that is,
4983:such that
4582:These are
3172:singletons
2937:convex sum
2433:dispersion
2382:such that
1790:, and the
1760:univariate
383:experiment
367:statistics
149:Experiment
96:Randomness
42:statistics
12535:Logistic
12302:posterior
12228:Rank sum
11976:Jackknife
11971:Bootstrap
11789:Bootstrap
11724:Parameter
11673:Statistic
11468:Statistic
11380:Run chart
11365:Pie chart
11360:Histogram
11350:Fan chart
11325:Bar chart
11207:L-moments
11094:Geometric
10962:combinant
10455:Dirichlet
10436:Dirichlet
10346:-Gaussian
10321:-Logistic
10158:Holtsmark
10130:Gaussian
10117:Fisher's
10100:real line
9602:Geometric
9582:Delaporte
9487:Bernoulli
9464:Discrete
9402:EMS Press
9319:cite book
9231:cite book
9223:927509011
8989:cite book
8871:122501973
8863:0020-739X
8813:710149819
8661:. Wiley.
8583:cite book
8575:262680588
8512:473463742
8476:190785258
8443:161828328
8407:Citations
8359:Histogram
8311:frequency
8195:Ψ
8169:∫
8148:≤
8142:≤
8124:Born rule
8047:(inverse
8045:precision
7997:R-squared
7752:power law
7641:λ
7615:−
7606:
7598:λ
7591:−
7451:−
7442:
7434:λ
7427:−
7364:−
7355:
7347:λ
7340:−
7328:⇔
7312:−
7303:
7291:λ
7288:−
7285:⇔
7272:−
7258:λ
7255:−
7247:⇔
7226:λ
7223:−
7215:−
7209:⇔
7156:λ
7153:−
7145:−
7098:≤
7051:≤
6897:−
6882:≥
6773:≥
6564:∞
6561:→
6529:
6364:≪
6351:≪
6254:→
6233:γ
6122:−
6099:∗
6003:∗
5942:∈
5936:ω
5927:∣
5924:Ω
5921:∈
5918:ω
5831:Ω
5637:∞
5634:−
5630:∫
5617:≤
5488:∫
5475:∈
5387:≤
5381:≤
5281:∫
5266:≤
5260:≤
5144:⊂
5100:∞
5088:→
4867:ω
4853:Ω
4828:∑
4818:ω
4770:…
4665:∑
4649:Ω
4633:∑
4615:Ω
4605:⋃
4568:…
4518:ω
4506:ω
4476:−
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4436:…
4337:ω
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4317:∈
4314:ω
4310:∑
4300:ω
4297:−
4288:δ
4279:∫
4272:ω
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4251:∑
4219:∫
4206:∈
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4059:ω
4055:δ
4048:ω
4034:∈
4031:ω
4027:∑
3979:ω
3975:δ
3968:ω
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3951:ω
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3853:∈
3804:ω
3782:ω
3778:δ
3757:ω
3713:ω
3699:≤
3696:ω
3692:∑
3679:≤
3623:. When a
3578:⋯
3356:∈
3301:ω
3281:∩
3275:∈
3272:ω
3268:∑
3255:∈
2899:→
2852:−
2828:≤
2769:∞
2766:→
2718:∞
2715:−
2712:→
2677:≤
2662:≤
2553:≤
1884:∈
1696:∈
1646:⊂
1485:∈
1467:∑
1441:⋃
1437:∈
1380:∈
1374:∀
1367:≤
1358:∈
1320:∈
1314:∀
1307:≥
1298:∈
1213:∈
1074:⊆
1042:Ï-algebra
996:→
986::
922:∞
919:−
875:for some
461:Ω
142:Singleton
13249:Category
12942:Survival
12819:Johansen
12542:Binomial
12497:Isotonic
12084:(normal)
11729:location
11536:Blocking
11491:Sampling
11370:QâQ plot
11335:Box plot
11317:Graphics
11212:Skewness
11202:Kurtosis
11174:Variance
11104:Heronian
11099:Harmonic
10957:cumulant
10927:L-moment
10922:kurtosis
10917:skewness
10907:variance
10788:Category
10720:Circular
10713:Families
10698:Singular
10677:singular
10441:Negative
10388:discrete
10354:-Weibull
10312:-Weibull
10196:Logistic
10080:Discrete
10050:Rayleigh
10030:Nakagami
9953:-squared
9927:Gompertz
9776:interval
9512:Negative
9497:Binomial
9367:25059432
9278:18669309
9190:71008143
8981:43953136
8708:14668369
8327:See also
8307:forecast
8272:such as
8111:used in
8049:variance
7738:normally
6795:so that
6766:if
6740:if
6630:. These
5960:satisfy
5404:integral
4108:, where
2925:discrete
2468:Kurtosis
2456:Skewness
2448:Symmetry
2428:Variance
2359:Quantile
1805:and the
1724:discrete
1591:, where
688:”
680:“
663:”
655:“
638:”
630:“
517:discrete
485:outcomes
393:and the
379:outcomes
375:function
223:Variance
13275:Commons
13222:Kriging
13107:Process
13064:studies
12923:Wavelet
12756:General
11923:Plug-in
11717:L space
11496:Cluster
11197:Moments
11015:Outline
10798:Commons
10770:Wrapped
10765:Tweedie
10760:Pearson
10755:Mixture
10662:Bingham
10561:Complex
10551:Inverse
10541:Wishart
10534:Inverse
10521:Matrix
10495:Inverse
10411:(joint)
10330:-Erlang
10184:Laplace
10075:Weibull
9932:Shifted
9915:Inverse
9900:Fréchet
9823:Inverse
9758:Uniform
9678:Arcsine
9637:Skellam
9632:Poisson
9555:support
9529:Soliton
9482:Benford
9475:support
9404:, 2001
9336:Sources
9080:Bibcode
8299:predict
8281:Fitting
8051:) of a
7694:numbers
7684:to the
7469:and if
5987:is the
5817:from a
5782:In the
5543:uniform
5065:has an
3238:) sum:
2921:mixture
2176:Support
1925:occurs.
781:exactly
493:vectors
479:is the
428:= tails
421:= heads
403:subsets
381:for an
137:Outcome
13144:Census
12734:Normal
12682:Manova
12502:Robust
12252:2-way
12244:1-way
12082:-test
11753:
11330:Biplot
11121:Median
11114:Lehmer
11056:Center
10704:Cantor
10546:Normal
10377:Mixed
10303:-Gamma
10229:Stable
10179:Landau
10153:Gumbel
10107:Cauchy
10035:Pareto
9847:Erlang
9828:Scaled
9783:Benini
9622:Panjer
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8055:, etc.
8043:, the
7489:has a
6324:plasma
5964:, the
5675:where
5551:others
5549:, and
5539:normal
4402:, let
3769:, let
3625:sample
3615:, the
3611:, the
3607:, the
3603:, the
3319:where
2961:(pmf)
2931:and a
2343:Median
2334:: the
2117:sample
1786:, the
1013:whose
951:
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925:
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884:
863:
838:
755:
620:
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575:
543:
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464:
458:
399:events
387:random
84:System
72:Axioms
12768:Trend
12297:prior
12239:anova
12128:-test
12102:-test
12094:-test
12001:Power
11946:Pivot
11739:shape
11734:scale
11184:Shape
11164:Range
11109:Heinz
11084:Cubic
11020:Index
10953:(pgf)
10942:(mgf)
10866:(cdf)
10860:(pdf)
10854:(pmf)
10426:Ewens
10252:Voigt
10224:Slash
10005:Lomax
10000:Log-t
9905:Gamma
9852:Hyper
9842:Davis
9837:Dagum
9693:Bates
9683:ARGUS
9567:Borel
9274:S2CID
9105:(PDF)
9028:(PDF)
8867:S2CID
8704:S2CID
8385:Lists
8086:of a
6223:balls
5864:to a
5221:over
4082:as a
3897:. If
3836:with
2927:, an
2923:of a
2801:; and
2152:) or
1937:that
1852:Event
1040:is a
908:from
610:event
117:Event
13001:Test
12201:Sign
12053:Wald
11126:Mode
11064:Mean
10902:mean
10675:and
10633:Kent
10060:Rice
9975:LĂ©vy
9803:Burr
9733:PERT
9698:Beta
9647:Zeta
9539:Zipf
9456:list
9363:PMID
9325:link
9307:OCLC
9297:ISBN
9264:ISBN
9237:link
9219:OCLC
9209:ISBN
9186:OCLC
9176:ISBN
9152:ISBN
9039:2019
8995:link
8977:OCLC
8967:ISBN
8890:ISBN
8859:ISSN
8809:OCLC
8799:ISBN
8780:see
8762:ISBN
8663:ISBN
8589:link
8571:OCLC
8561:ISBN
8508:OCLC
8498:ISBN
8472:OCLC
8462:ISBN
8439:OCLC
8429:ISBN
8309:the
8301:the
8103:The
8065:and
8026:and
7979:mean
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6694:<
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5790:, a
5776:and
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3484:for
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2230:Tail
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1141:and
854:<
519:and
369:, a
365:and
12181:BIC
12176:AIC
10511:LKJ
9808:Chi
9355:doi
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8851:doi
8696:doi
8039:or
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6526:sin
6322:in
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6188:or
6046:on
6022:of
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5040:An
4802:as
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2150:pdf
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563:to
561:â1â
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6858:X
6855:(
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