1506:
60:
48:
15915:
1201:
13349:
18809:
4751:
12698:
18819:
5861:
4177:
12281:
5613:
10660:
11858:
6630:
3398:
14510:
4746:{\displaystyle {\begin{aligned}\mu _{1}&=0,\\\mu _{2}&=np(1-p),\\\mu _{3}&=np(1-p)(1-2p),\\\mu _{4}&=np(1-p)(1+(3n-6)p(1-p)),\\\mu _{5}&=np(1-p)(1-2p)(1+(10n-12)p(1-p)),\\\mu _{6}&=np(1-p)(1-30p(1-p)(1-4p(1-p))+5np(1-p)(5-26p(1-p))+15n^{2}p^{2}(1-p)^{2}).\end{aligned}}}
8248:
13639:
17023:
11063:
13227:
5350:
8521:
3605:
12693:{\displaystyle {\begin{aligned}\Pr&={\binom {n}{m}}p^{m}q^{m}\left(\sum _{k=m}^{n}{\binom {n-m}{k-m}}p^{k-m}(1-p)^{n-k}(1-q)^{k-m}\right)\\&={\binom {n}{m}}(pq)^{m}\left(\sum _{k=m}^{n}{\binom {n-m}{k-m}}\left(p(1-q)\right)^{k-m}(1-p)^{n-k}\right)\end{aligned}}}
14612:
4893:
15775:
3993:
14773:
must be greater than or equal to 5. However, the specific number varies from source to source, and depends on how good an approximation one wants. In particular, if one uses 9 instead of 5, the rule implies the results stated in the previous
12941:
10198:
8367:
15113:
14372:
13851:
14707:
5856:{\displaystyle {\text{mode}}={\begin{cases}\lfloor (n+1)\,p\rfloor &{\text{if }}(n+1)p{\text{ is 0 or a noninteger}},\\(n+1)\,p\ {\text{ and }}\ (n+1)\,p-1&{\text{if }}(n+1)p\in \{1,\dots ,n\},\\n&{\text{if }}(n+1)p=n+1.\end{cases}}}
12194:
5008:
3098:
7392:
10484:
9730:
14165:
14046:
11593:
9854:
7886:
6404:
3154:
11996:
8041:
7742:
9046:
898:
14409:
14964:
2563:
9145:
9953:
8780:
8104:
2928:
13501:
10750:
15638:
13748:
6392:
9359:
6879:
1897:
16682:
13016:
8889:
5187:
438:
6998:
4169:
13021:
12286:
11598:
8394:
7094:
5129:
3420:
8960:
1993:
8699:
5048:
2281:
9293:
4074:
15277:
10755:
9448:
6409:
4767:
4182:
3159:
13930:
5953:
10473:
15461:
10407:
14521:
8596:
13452:
4762:
6799:
364:
15660:
3836:
745:
13654:
A stronger rule states that the normal approximation is appropriate only if everything within 3 standard deviations of its mean is within the range of possible values; that is, only if
7619:
2644:
12741:
10107:
9601:
9508:
7175:
1152:
8259:
16472:
15021:
14280:
13759:
5565:
5539:
5475:
3711:
2388:
636:
589:
290:
160:
7541:
3828:
11199:
802:
3772:
2788:
14623:
12253:
9230:
5436:
511:
15216:
people out of a large population and ask them whether they agree with a certain statement. The proportion of people who agree will of course depend on the sample. If groups of
10324:
9194:
5397:
3127:
13004:
7462:
7425:
7292:
7252:
7208:
540:
10277:
1025:
965:
13274:
10021:
12004:
11581:
11540:
2183:
2123:
2084:
1722:
14868:
14200:
6275:
4904:
694:
14272:
10655:{\displaystyle {\frac {{\widehat {p\,}}+{\frac {z^{2}}{2n}}+z{\sqrt {{\frac {{\widehat {p\,}}(1-{\widehat {p\,}})}{n}}+{\frac {z^{2}}{4n^{2}}}}}}{1+{\frac {z^{2}}{n}}}}}
2948:
10086:
1090:
15851:
to generate samples uniformly between 0 and 1, one can transform the calculated samples into discrete numbers by using the probabilities calculated in the first step.
10047:
8092:
7305:
6715:
6674:
10344:
9982:
9563:
9539:
6168:
14836:
14771:
14401:
14230:
9641:
6750:
6142:
6107:
6046:
102:
12733:
232:
199:
14057:
13938:
6011:
15013:
6240:
6194:
6072:
5982:
14801:
14736:
11853:{\displaystyle {\begin{aligned}\Pr&=\sum _{k=m}^{n}\Pr\Pr\\&=\sum _{k=m}^{n}{\binom {n}{k}}{\binom {k}{m}}p^{k}q^{m}(1-p)^{n-k}(1-q)^{k-m}\end{aligned}}}
6941:
6918:
6625:{\displaystyle {\begin{aligned}k>(n+1)p-1\Rightarrow f(k+1)<f(k)\\k=(n+1)p-1\Rightarrow f(k+1)=f(k)\\k<(n+1)p-1\Rightarrow f(k+1)>f(k)\end{aligned}}}
474:
14987:
12273:
9621:
9470:
6214:
5173:
5149:
3393:{\displaystyle {\begin{aligned}F(k;n,p)&=\Pr(X\leq k)\\&=I_{1-p}(n-k,k+1)\\&=(n-k){n \choose k}\int _{0}^{1-p}t^{n-k-1}(1-t)^{k}\,dt.\end{aligned}}}
1173:
13343:
9379:, the actual distribution of the mean is significantly nonnormal. Because of this problem several methods to estimate confidence intervals have been proposed.
9762:
7796:
17166:
11866:
7929:
7641:
17187:
17101:
17044:
15220:
people were sampled repeatedly and truly randomly, the proportions would follow an approximate normal distribution with mean equal to the true proportion
10673:
17467:
8987:
10097:
9752:
14505:{\displaystyle {\frac {\sqrt {n}}{3}}>{\sqrt {\frac {1-p}{p}}}>0\quad {\text{and}}\quad {\frac {\sqrt {n}}{3}}>{\sqrt {\frac {p}{1-p}}}>0.}
10209:
1492:
15502:
The binomial distribution and beta distribution are different views of the same model of repeated
Bernoulli trials. The binomial distribution is the
815:
15144: ≤ 8.5). The addition of 0.5 is the continuity correction; the uncorrected normal approximation gives considerably less accurate results.
14876:
59:
8243:{\displaystyle F(k;n,p)\geq {\frac {1}{\sqrt {8n{\tfrac {k}{n}}(1-{\tfrac {k}{n}})}}}\exp \left(-nD\left({\frac {k}{n}}\parallel p\right)\right),}
2430:
16301:
Hamza, K. (1995). "The smallest uniform upper bound on the distance between the mean and the median of the binomial and
Poisson distributions".
13634:{\displaystyle {\frac {|1-2p|}{\sqrt {np(1-p)}}}={\frac {1}{\sqrt {n}}}\left|{\sqrt {\frac {1-p}{p}}}-{\sqrt {\frac {p}{1-p}}}\,\right|<0.3.}
9054:
16744:
11058:{\displaystyle {\begin{aligned}\operatorname {P} (Z=k)&=\sum _{i=0}^{k}\left\left\\&={\binom {n+m}{k}}p^{k}(1-p)^{n+m-k}\end{aligned}}}
9869:
9633:
8711:
9196:
which sometimes is unrealistic and undesirable. In such cases there are various alternative estimators. One way is to use the Bayes estimator
2823:
16762:
47:
16461:
15557:
971:
17596:
13663:
13222:{\displaystyle {\begin{aligned}\Pr&={\binom {n}{m}}(pq)^{m}(p-pq+1-p)^{n-m}\\&={\binom {n}{m}}(pq)^{m}(1-pq)^{n-m}\end{aligned}}}
6283:
16869:
18822:
18079:
9312:
6804:
5345:{\displaystyle \operatorname {E} \leq \left({\frac {c}{\log(c/(np)+1)}}\right)^{c}\leq (np)^{c}\exp \left({\frac {c^{2}}{2np}}\right).}
1777:
8823:
18858:
17987:
377:
18774:
6953:
17430:
4094:
18640:
17852:
17611:
17460:
9370:
8516:{\displaystyle F(k;n,{\tfrac {1}{2}})\geq {\frac {1}{15}}\exp \left(-16n\left({\frac {1}{2}}-{\frac {k}{n}}\right)^{2}\right).\!}
2029:
trials result in “failure“. Since the trials are independent with probabilities remaining constant between them, any sequence of
17080:
3600:{\displaystyle F(k;n,p)=F_{F{\text{-distribution}}}\left(x={\frac {1-p}{p}}{\frac {k+1}{n-k}};d_{1}=2(n-k),d_{2}=2(k+1)\right).}
18853:
18848:
18535:
18299:
10680:
does not mean perfectly accurate; rather, it indicates that the estimates will not be less conservative than the true value.)
1485:
7010:
17973:
17318:
16852:
15996:
5057:
4079:
This similarly follows from the fact that the variance of a sum of independent random variables is the sum of the variances.
17200:
Novak S.Y. (2011) Extreme value methods with applications to finance. London: CRC/ Chapman & Hall/Taylor & Francis.
8897:
1918:
18294:
18238:
18136:
17898:
17536:
15847:. (These probabilities should sum to a value close to one, in order to encompass the entire sample space.) Then by using a
8666:
5016:
3145:
2047:
failures) has the same probability of being achieved (regardless of positions of successes within the sequence). There are
1664:. If the sampling is carried out without replacement, the draws are not independent and so the resulting distribution is a
9238:
4008:
2730:
outcome (that is, the most likely, although this can still be unlikely overall) of the
Bernoulli trials and is called the
2188:
18580:
18314:
18167:
17842:
17586:
17205:
15823:
samples from a binomial distribution is to use an inversion algorithm. To do so, one must calculate the probability that
15230:
9401:
8614:
5051:
18044:
18812:
18484:
18460:
18039:
17453:
17227:
16144:
14607:{\displaystyle {\frac {\sqrt {n}}{3}}>{\sqrt {\frac {1-p}{p}}}-{\sqrt {\frac {p}{1-p}}}>-{\frac {\sqrt {n}}{3}};}
13863:
5875:
10412:
4888:{\displaystyle {\begin{aligned}\operatorname {E} &=np,\\\operatorname {E} &=np(1-p)+n^{2}p^{2},\end{aligned}}}
18681:
18558:
18519:
18491:
18465:
18383:
18309:
17732:
17480:
17393:
17124:
17007:
16982:
16444:
15408:
10349:
1575:
1478:
1466:
1425:
8556:
6893:
for a binomial distribution, and it may even be non-unique. However, several special results have been established:
18669:
18635:
18501:
18496:
18341:
18149:
17847:
17601:
15770:{\displaystyle P(p;\alpha ,\beta )={\frac {p^{\alpha -1}(1-p)^{\beta -1}}{\operatorname {Beta} (\alpha ,\beta )}}.}
13393:
11144:
3988:{\displaystyle \operatorname {E} =\operatorname {E} =\operatorname {E} +\cdots +\operatorname {E} =p+\cdots +p=np.}
3404:
2939:
370:
13376:
is large enough, then the skew of the distribution is not too great. In this case a reasonable approximation to B(
18843:
18419:
18332:
18304:
18213:
18162:
18034:
17817:
17782:
16181:
D. Ahle, Thomas (2022), "Sharp and Simple Bounds for the raw
Moments of the Binomial and Poisson Distributions",
13754:
This 3-standard-deviation rule is equivalent to the following conditions, which also imply the first rule above.
13492:
the normal approximation is adequate if the absolute value of the skewness is strictly less than 0.3; that is, if
12936:{\displaystyle \Pr={\binom {n}{m}}(pq)^{m}\left(\sum _{i=0}^{n-m}{\binom {n-m}{i}}(p-pq)^{i}(1-p)^{n-m-i}\right)}
10193:{\displaystyle \sin ^{2}\left(\arcsin \left({\sqrt {\widehat {p\,}}}\right)\pm {\frac {z}{2{\sqrt {n}}}}\right).}
7904:
6755:
1585:
1356:
1292:
1031:
304:
16937:
8362:{\displaystyle F(k;n,p)\geq {\frac {1}{\sqrt {2n}}}\exp \left(-nD\left({\frac {k}{n}}\parallel p\right)\right).}
707:
18433:
18350:
18187:
18111:
17934:
17812:
17787:
17651:
17646:
17641:
15938:
15108:{\displaystyle n>9\left({\frac {1-p}{p}}\right)\quad {\text{and}}\quad n>9\left({\frac {p}{1-p}}\right).}
14367:{\displaystyle n>9\left({\frac {1-p}{p}}\right)\quad {\text{and}}\quad n>9\left({\frac {p}{1-p}}\right).}
13846:{\displaystyle n>9\left({\frac {1-p}{p}}\right)\quad {\text{and}}\quad n>9\left({\frac {p}{1-p}}\right).}
8606:
7550:
2578:
1404:
1265:
31:
17:
9568:
9475:
8618:
7114:
1103:
18749:
18615:
18323:
18172:
18104:
18089:
17982:
17956:
17888:
17727:
17621:
17616:
17558:
17543:
15848:
15806:
15387:
15148:
11209:
8813:
8794:
8786:
5152:
5544:
5503:
5441:
3672:
2310:
594:
553:
245:
115:
18585:
18575:
18266:
18192:
17893:
17752:
16589:
14702:{\displaystyle \left|{\sqrt {\frac {1-p}{p}}}-{\sqrt {\frac {p}{1-p}}}\,\right|<{\frac {\sqrt {n}}{3}}.}
13357:
9514:
8602:
7484:
3781:
18645:
11150:
5176:
2817:
comes up heads with probability 0.3 when tossed. The probability of seeing exactly 4 heads in 6 tosses is
2286:
In creating reference tables for binomial distribution probability, usually, the table is filled in up to
758:
18630:
18625:
18570:
18506:
18450:
18271:
18258:
18049:
17994:
17946:
17737:
17666:
17531:
8095:
7621:, these bounds can also be seen as bounds for the upper tail of the cumulative distribution function for
3731:
3134:
2791:
2740:
1665:
12202:
10231:
th quantile of the standard normal distribution', rather than being a shorthand for 'the (1 −
9199:
8537:
5402:
487:
18764:
18540:
18359:
18141:
18094:
17963:
17939:
17919:
17762:
17636:
17516:
17243:
16328:
Nowakowski, Sz. (2021). "Uniqueness of a Median of a
Binomial Distribution with Rational Probability".
15943:
10282:
9303:
9161:
5360:
3106:
1260:
914:
12949:
7434:
7397:
7264:
7224:
7180:
516:
441:
18769:
18553:
18514:
18388:
18225:
18069:
18014:
17912:
17876:
17747:
17712:
17333:
Mandelbrot, B. B., Fisher, A. J., & Calvet, L. E. (1997). A multifractal model of asset returns.
16901:
Katz, D.; et al. (1978). "Obtaining confidence intervals for the risk ratio in cohort studies".
16733:
15503:
13353:
12189:{\displaystyle \Pr=\sum _{k=m}^{n}{\binom {n}{m}}{\binom {n-m}{k-m}}p^{k}q^{m}(1-p)^{n-k}(1-q)^{k-m}}
10241:
8626:
7632:
1768:
1376:
978:
297:
16606:
5630:
5003:{\displaystyle \operatorname {E} =\sum _{k=0}^{c}\left\{{c \atop k}\right\}n^{\underline {k}}p^{k},}
921:
18455:
18243:
18009:
17968:
17883:
17837:
17777:
17742:
17631:
17526:
17476:
16707:
16258:
15933:
15812:
13647:
13235:
11584:
9987:
3093:{\displaystyle F(k;n,p)=\Pr(X\leq k)=\sum _{i=0}^{\lfloor k\rfloor }{n \choose i}p^{i}(1-p)^{n-i},}
1646:
1582:
1435:
1430:
1319:
1304:
17150:
16816:
16793:
15899: = 1/2, tabulating the corresponding binomial coefficients in what is now recognized as
11545:
11504:
10227:
has a slightly different interpretation in the formula below: it has its ordinary meaning of 'the
1705:
1652:
The binomial distribution is frequently used to model the number of successes in a sample of size
18754:
18696:
18367:
18154:
18064:
18019:
18004:
17822:
17772:
17767:
17568:
17548:
17065:
15161:
14841:
14173:
7387:{\displaystyle {\frac {1}{2}}{\bigl (}n-1{\bigr )}\leq m\leq {\frac {1}{2}}{\bigl (}n+1{\bigr )}}
6248:
2149:
2089:
2050:
1593:
1414:
1285:
649:
17924:
17415:
15155:
are very onerous); historically, it was the first use of the normal distribution, introduced in
14235:
10238:
Secondly, this formula does not use a plus-minus to define the two bounds. Instead, one may use
8966:
8702:
18620:
18608:
18597:
18479:
18375:
18182:
17626:
17606:
17511:
16734:"Confidence intervals for a binomial proportion: comparison of methods and software evaluation"
16601:
16044:
16021:
15206:
15182:
13287:
11084:
Bernoulli distributed random variables. So the sum of two
Binomial distributed random variable
10052:
9725:{\displaystyle {\widehat {p\,}}\pm z{\sqrt {\frac {{\widehat {p\,}}(1-{\widehat {p\,}})}{n}}}.}
8802:
1638:
1309:
1038:
17372:
16974:
16635:
16436:
16384:
Arratia, R.; Gordon, L. (1989). "Tutorial on large deviations for the binomial distribution".
15327: ≤ 0.05 such that np ≤ 1, or if n > 50 and p < 0.1 such that np < 5, or if
10026:
8056:
6679:
6638:
18744:
18701:
18545:
18220:
18074:
18054:
17951:
16884:
16536:
16490:
16093:
15958:
15952:
15816:
15490:
15348:
15166:
15121:
14160:{\displaystyle np>3{\sqrt {np(1-p)}}\quad {\text{and}}\quad n(1-p)>3{\sqrt {np(1-p)}}.}
14041:{\displaystyle np-3{\sqrt {np(1-p)}}>0\quad {\text{and}}\quad np+3{\sqrt {np(1-p)}}<n.}
13458:
10329:
9961:
9736:
9548:
9524:
8817:
6147:
1597:
1505:
1450:
1409:
1280:
238:
108:
16428:
16013:
15338:
Concerning the accuracy of
Poisson approximation, see Novak, ch. 4, and references therein.
15151:, is a huge time-saver when undertaking calculations by hand (exact calculations with large
14806:
14741:
14380:
14205:
6720:
6112:
6077:
6016:
72:
18794:
18789:
18784:
18779:
18716:
18686:
18565:
18208:
18099:
17702:
17661:
17656:
17553:
15900:
15482:
15373:
15288:
12706:
9382:
In the equations for confidence intervals below, the variables have the following meaning:
8974:
8806:
8798:
8634:
8610:
1999:
1523:
1440:
1334:
1227:
205:
166:
17999:
5987:
8:
18728:
18253:
18233:
18203:
18177:
18131:
18059:
17871:
17807:
15949:
15928:
15470:
14992:
13385:
9299:
7481:, upper bounds can be derived for the lower tail of the cumulative distribution function
6219:
6173:
6051:
5961:
1589:
1399:
1341:
1329:
1324:
808:
17436:
14783:
14718:
9849:{\displaystyle {\tilde {p}}\pm z{\sqrt {\frac {{\tilde {p}}(1-{\tilde {p}})}{n+z^{2}}}}}
7881:{\displaystyle F(k;n,p)\leq \exp \left(-nD\left({\frac {k}{n}}\parallel p\right)\right)}
6923:
6900:
456:
18759:
18248:
18029:
18024:
17929:
17866:
17861:
17717:
17707:
17591:
17291:
16918:
16785:
16658:
16518:
16409:
16363:
16337:
16287:
16190:
16164:
16126:
16075:
16037:
15920:
15651:
15647:
14972:
12258:
9606:
9455:
8970:
8660:
8638:
8622:
6199:
5158:
5134:
1555:
1386:
1275:
1215:
1192:
1158:
1096:
17274:
Kachitvichyanukul, V.; Schmeiser, B. W. (1988). "Binomial random variate generation".
16145:"A probabilistic approach to the moments of binomial random variables and application"
11991:{\displaystyle {\tbinom {n}{k}}{\tbinom {k}{m}}={\tbinom {n}{m}}{\tbinom {n-m}{k-m}},}
10214:
The notation in the formula below differs from the previous formulas in two respects:
8036:{\displaystyle D(a\parallel p)=(a)\log {\frac {a}{p}}+(1-a)\log {\frac {1-a}{1-p}}.\!}
7737:{\displaystyle F(k;n,p)\leq \exp \left(-2n\left(p-{\frac {k}{n}}\right)^{2}\right),\!}
18657:
18084:
17827:
17757:
17722:
17671:
17389:
17314:
17223:
17201:
17181:
17120:
17095:
17038:
17003:
16978:
16967:
16848:
16522:
16510:
16440:
16429:
16413:
16401:
16367:
16355:
16314:
16168:
16014:
15992:
15914:
15643:
15156:
8653:
8094:, known as anti-concentration bounds. By approximating the binomial coefficient with
5485:
2731:
1627:
1445:
1351:
1250:
546:
17410:
17295:
16636:"Approximate is better than 'exact' for interval estimation of binomial proportions"
15477: 1. This result is sometimes loosely stated by saying that the distribution of
17832:
17506:
17445:
17283:
16910:
16777:
16650:
16611:
16565:
16502:
16393:
16347:
16310:
16283:
16274:
Kaas, R.; Buhrman, J.M. (1980). "Mean, Median and Mode in
Binomial Distributions".
16200:
16156:
16118:
16067:
16058:
Jowett, G. H. (1963). "The
Relationship Between the Binomial and F Distributions".
15319:
is sufficiently small. According to rules of thumb, this approximation is good if
13323:
13007:
11217:
9542:
9041:{\displaystyle \operatorname {Beta} (\alpha ={\frac {1}{2}},\beta ={\frac {1}{2}})}
1657:
1270:
1200:
16554:"On the estimation of binomial success probability with zero occurrence in sample"
16160:
16107:"Closed-Form Expressions for the Moments of the Binomial Probability Distribution"
16842:
15963:
15860:
15780:
Given a uniform prior, the posterior distribution for the probability of success
15544:, the beta distribution and the binomial distribution are related by a factor of
15190:
13457:
and this basic approximation can be improved in a simple way by using a suitable
10683:
The Wald method, although commonly recommended in textbooks, is the most biased.
8657:
8645:
2133:
trials. The binomial distribution is concerned with the probability of obtaining
1690:
1623:
1346:
1297:
751:
17024:"6.4: Normal Approximation to the Binomial Distribution - Statistics LibreTexts"
16570:
16553:
15295:
converges to a finite limit. Therefore, the
Poisson distribution with parameter
30:"Binomial model" redirects here. For the binomial model in options pricing, see
17905:
16506:
16224:
15820:
15377:
8981:
7787:
5568:
4088:
3716:
This follows from the linearity of the expected value along with the fact that
3659:
3408:
1702:
1361:
902:
893:{\displaystyle {\frac {1}{2}}\log _{2}(2\pi enpq)+O\left({\frac {1}{n}}\right)}
449:
16204:
1676:, the binomial distribution remains a good approximation, and is widely used.
18837:
18528:
18276:
17563:
16514:
16359:
15892:
14959:{\displaystyle np\geq 9>9(1-p)\quad {\text{and}}\quad n(1-p)\geq 9>9p.}
13470:
1642:
1234:
16615:
13344:
Binomial proportion confidence interval § Normal approximation interval
17421:
16540:
16239:
Neumann, P. (1966). "Über den Median der Binomial- and Poissonverteilung".
3610:
Some closed-form bounds for the cumulative distribution function are given
2558:{\displaystyle {\frac {f(k+1,n,p)}{f(k,n,p)}}={\frac {(n-k)p}{(k+1)(1-p)}}}
2002:, hence the distribution's name. The formula can be understood as follows:
1529:
1461:
1371:
1255:
17425:
16405:
16257:
Lord, Nick. (July 2010). "Binomial averages when the mean is an integer",
16219:
10326:
to get the upper bound. For example: for a 95% confidence level the error
9140:{\displaystyle {\widehat {p}}_{Jeffreys}={\frac {x+{\frac {1}{2}}}{n+1}}.}
3774:
are identical (and independent) Bernoulli random variables with parameter
3658:
the probability of each experiment yielding a successful result, then the
17261:
16351:
9948:{\displaystyle {\tilde {p}}={\frac {n_{1}+{\frac {1}{2}}z^{2}}{n+z^{2}}}}
8775:{\displaystyle {\widehat {p}}_{b}={\frac {x+\alpha }{n+\alpha +\beta }}.}
1381:
1222:
1210:
17287:
16130:
16106:
13348:
2923:{\displaystyle f(4,6,0.3)={\binom {6}{4}}0.3^{4}(1-0.3)^{6-4}=0.059535.}
16922:
16789:
16662:
16397:
16079:
8969:
should just lead to the standard estimator.) This method is called the
1559:
1239:
1185:
906:
16763:"Probable inference, the law of succession, and statistical inference"
16539:), Jeffreys prior for binomial likelihood, URL (version: 2019-03-04):
16122:
14515:
Subtracting the second set of inequalities from the first one yields:
2137:
of these sequences, meaning the probability of obtaining one of them (
15819:
is a binomial distribution are well-established. One way to generate
15633:{\displaystyle \operatorname {Beta} (p;\alpha ;\beta )=(n+1)B(k;n;p)}
14202:, we can apply the square power and divide by the respective factors
10098:
Binomial proportion confidence interval § Arcsine transformation
9753:
Binomial proportion confidence interval § Agresti–Coull interval
2814:
17119:(1 ed.). Singapore: Educational publishing house. p. 348.
17002:(1 ed.). Singapore: Educational Publishing House. p. 350.
16914:
16781:
16654:
16071:
10210:
Binomial proportion confidence interval § Wilson score interval
17146:
17061:
16841:
Dekking, F.M.; Kraaikamp, C.; Lopohaa, H.P.; Meester, L.E. (2005).
16342:
16195:
15486:
15474:
15015:, to deduce the alternative form of the 3-standard-deviation rule:
13743:{\displaystyle \mu \pm 3\sigma =np\pm 3{\sqrt {np(1-p)}}\in (0,n).}
9510:
7104:
6387:{\displaystyle {\frac {f(k+1)}{f(k)}}={\frac {(n-k)p}{(k+1)(1-p)}}}
5175:. A simple bound follows by bounding the Binomial moments via the
3999:
1366:
700:
642:
17335:
3.2 The Binomial Measure is the Simplest Example of a Multifractal
14403:. On the other hand, apply again the square root and divide by 3,
11417:
and taking the balls that hit and throwing them to another basket
8046:
Asymptotically, this bound is reasonably tight; see for details.
6920:
is an integer, then the mean, median, and mode coincide and equal
16241:
Wissenschaftliche Zeitschrift der Technischen Universität Dresden
9354:{\displaystyle {\widehat {p}}_{\text{rule of 3}}={\frac {3}{n}}.}
6874:{\displaystyle \lfloor (n+1)p-1\rfloor +1=\lfloor (n+1)p\rfloor }
1892:{\displaystyle f(k,n,p)=\Pr(X=k)={\binom {n}{k}}p^{k}(1-p)^{n-k}}
17373:"Binomial Distribution—Success or Failure, How Likely Are They?"
8884:{\displaystyle \operatorname {Beta} (\alpha =1,\beta =1)=U(0,1)}
1626:
or Bernoulli experiment, and a sequence of outcomes is called a
17388:(Third ed.). Boston: Allyn & Bacon. pp. 185–192.
17167:"The Connection Between the Poisson and Binomial Distributions"
10720:) are independent binomial variables with the same probability
9518:
6890:
3724:
identical Bernoulli random variables, each with expected value
480:
17431:
Querying the binomial probability distribution in WolframAlpha
16840:
15136:
has a distribution given by the normal approximation, then Pr(
433:{\displaystyle I_{q}(n-\lfloor k\rfloor ,1+\lfloor k\rfloor )}
17435:
Confidence (credible) intervals for binomial probability, p:
17066:"7.2.4. Does the proportion of defectives meet requirements?"
16491:"Estimating the Parameters of the Beta-Binomial Distribution"
11235:
This result was first derived by Katz and coauthors in 1978.
11128:). This can also be proven directly using the addition rule.
6993:{\displaystyle \lfloor np\rfloor \leq m\leq \lceil np\rceil }
17440:
16938:"Lectures on Probability Theory and Mathematical Statistics"
16588:
Brown, Lawrence D.; Cai, T. Tony; DasGupta, Anirban (2001),
9634:
Binomial proportion confidence interval § Wald interval
4164:{\displaystyle \mu _{c}=\operatorname {E} \left)^{c}\right]}
2695:
is an integer. In this case, there are two values for which
17142:
17081:"12.4 - Approximating the Binomial Distribution | STAT 414"
17057:
15989:
Audit Analytics: Data Science for the Accounting Profession
15291:
as the number of trials goes to infinity while the product
15224:
of agreement in the population and with standard deviation
9158:(e.g.: if x=0), then using the standard estimator leads to
7783:), but Hoeffding's bound evaluates to a positive constant.
5849:
5607:
correspondingly. These cases can be summarized as follows:
17273:
16537:
https://stats.stackexchange.com/users/105848/marko-lalovic
16016:
An Introduction to Probability Theory and Its Applications
15165:
in 1738. Nowadays, it can be seen as a consequence of the
5583:
is neither 0 nor 1, then the distribution has two modes: (
17384:
Neter, John; Wasserman, William; Whitmore, G. A. (1988).
15967:
2304:, the probability can be calculated by its complement as
2125:
counts the number of ways to choose the positions of the
1641:. The binomial distribution is the basis for the popular
10346: = 0.05, so one gets the lower bound by using
7089:{\displaystyle |m-np|\leq \min\{{\ln 2},\max\{p,1-p\}\}}
1622:). A single success/failure experiment is also called a
11391:
is a simple binomial random variable with distribution
5124:{\displaystyle n^{\underline {k}}=n(n-1)\cdots (n-k+1)}
17383:
17169:. 2023-03-13. Archived from the original on 2023-03-13
17083:. 2023-03-28. Archived from the original on 2023-03-28
17026:. 2023-05-29. Archived from the original on 2023-05-29
16964:
14377:
Notice that these conditions automatically imply that
13290:
is a special case of the binomial distribution, where
12946:
Notice that the sum (in the parentheses) above equals
11946:
11922:
11895:
11871:
11304:) is approximately normally distributed with mean log(
11230:
9579:
9486:
8955:{\displaystyle {\widehat {p}}_{b}={\frac {x+1}{n+2}}.}
8417:
8169:
8148:
5020:
2191:
2152:
2092:
2053:
1988:{\displaystyle {\binom {n}{k}}={\frac {n!}{k!(n-k)!}}}
17309:
Katz, Victor (2009). "14.3: Elementary Probability".
17220:
Information Theory, Inference and Learning Algorithms
15663:
15560:
15411:
15233:
15024:
14995:
14975:
14969:
We only have to divide now by the respective factors
14879:
14844:
14809:
14786:
14744:
14721:
14626:
14524:
14412:
14383:
14283:
14238:
14208:
14176:
14060:
13941:
13866:
13762:
13666:
13504:
13396:
13238:
13019:
12952:
12744:
12709:
12284:
12261:
12205:
12007:
11869:
11596:
11548:
11507:
11153:
10753:
10487:
10415:
10352:
10332:
10285:
10244:
10110:
10055:
10029:
9990:
9964:
9872:
9765:
9644:
9609:
9571:
9551:
9527:
9478:
9458:
9404:
9315:
9241:
9202:
9164:
9057:
8990:
8900:
8826:
8714:
8694:{\displaystyle \operatorname {Beta} (\alpha ,\beta )}
8669:
8559:
8397:
8262:
8107:
8059:
7932:
7799:
7644:
7553:
7487:
7437:
7400:
7308:
7267:
7227:
7183:
7117:
7013:
6956:
6926:
6903:
6807:
6758:
6723:
6682:
6641:
6407:
6286:
6251:
6222:
6202:
6176:
6150:
6115:
6080:
6054:
6019:
5990:
5964:
5878:
5616:
5547:
5506:
5444:
5405:
5363:
5190:
5161:
5137:
5060:
5043:{\displaystyle \textstyle \left\{{c \atop k}\right\}}
5019:
4907:
4765:
4180:
4097:
4011:
3839:
3784:
3734:
3675:
3423:
3157:
3109:
2951:
2826:
2743:
2581:
2433:
2313:
2276:{\textstyle \Pr(X=k)={\binom {n}{k}}p^{k}(1-p)^{n-k}}
1921:
1780:
1708:
1161:
1106:
1041:
981:
924:
818:
761:
710:
652:
597:
556:
519:
490:
459:
380:
307:
248:
208:
169:
118:
75:
17475:
16900:
15910:
11116:
Bernoulli distributed random variables, which means
9288:{\displaystyle {\widehat {p}}_{b}={\frac {1}{n+2}}.}
8550:
can be estimated using the proportion of successes:
7747:
which is however not very tight. In particular, for
4069:{\displaystyle \operatorname {Var} (X)=npq=np(1-p).}
16844:
A Modern Introduction of Probability and Statistics
15272:{\displaystyle \sigma ={\sqrt {\frac {p(1-p)}{n}}}}
11208:The binomial distribution is a special case of the
9443:{\displaystyle {\widehat {p\,}}={\frac {n_{1}}{n}}}
8388:, it is possible to make the denominator constant:
6889:In general, there is no single formula to find the
2933:
16966:
16870:"On the number of successes in independent trials"
16060:Journal of the Royal Statistical Society, Series D
16036:
15769:
15632:
15455:
15271:
15107:
15007:
14981:
14958:
14862:
14830:
14795:
14765:
14730:
14701:
14606:
14504:
14395:
14366:
14266:
14224:
14194:
14159:
14040:
13924:
13845:
13742:
13633:
13446:
13268:
13221:
12998:
12935:
12727:
12692:
12267:
12247:
12188:
11990:
11852:
11575:
11534:
11193:
11057:
10732:is again a binomial variable; its distribution is
10654:
10467:
10401:
10338:
10318:
10271:
10192:
10080:
10041:
10015:
9976:
9947:
9848:
9724:
9615:
9595:
9557:
9533:
9502:
9464:
9442:
9353:
9287:
9224:
9188:
9139:
9040:
8954:
8883:
8774:
8693:
8590:
8515:
8361:
8242:
8086:
8035:
7880:
7736:
7613:
7535:
7456:
7419:
7386:
7286:
7246:
7202:
7169:
7088:
6992:
6935:
6912:
6873:
6793:
6744:
6709:
6668:
6624:
6386:
6269:
6234:
6208:
6188:
6162:
6136:
6101:
6066:
6040:
6005:
5976:
5947:
5855:
5559:
5533:
5469:
5430:
5391:
5344:
5167:
5143:
5123:
5042:
5002:
4887:
4745:
4163:
4068:
3987:
3822:
3766:
3705:
3599:
3392:
3121:
3092:
2922:
2782:
2638:
2568:and comparing it to 1. There is always an integer
2557:
2382:
2275:
2177:
2117:
2078:
1987:
1891:
1716:
1696:follows the binomial distribution with parameters
1167:
1146:
1084:
1019:
959:
892:
796:
739:
688:
630:
583:
534:
505:
468:
432:
358:
284:
226:
193:
154:
96:
16739:. In Klinke, S.; Ahrend, P.; Richter, L. (eds.).
13925:{\displaystyle np\pm 3{\sqrt {np(1-p)}}\in (0,n)}
13159:
13146:
13065:
13052:
12863:
12842:
12782:
12769:
12606:
12577:
12523:
12510:
12414:
12385:
12330:
12317:
12104:
12075:
12066:
12053:
11764:
11751:
11742:
11729:
11001:
10980:
10906:
10885:
10828:
10815:
8512:
8032:
7905:relative entropy (or Kullback-Leibler divergence)
7733:
5948:{\displaystyle f(k)={\binom {n}{k}}p^{k}q^{n-k}.}
5910:
5897:
3305:
3292:
3043:
3030:
2870:
2857:
1938:
1925:
1845:
1832:
1763:independent Bernoulli trials (with the same rate
324:
311:
18835:
17186:: CS1 maint: bot: original URL status unknown (
17100:: CS1 maint: bot: original URL status unknown (
17043:: CS1 maint: bot: original URL status unknown (
16587:
16039:Introduction to Probability and Random Variables
15287:The binomial distribution converges towards the
13481:is far enough from the extremes of zero or one:
13461:. The basic approximation generally improves as
13024:
12745:
12289:
12008:
11677:
11647:
11601:
11203:
11145:smaller than the variance of a binomial variable
10468:{\displaystyle z=z_{1-\alpha /2}=z_{0.975}=1.96}
9298:Another method is to use the upper bound of the
7554:
7515:
7143:
7059:
7039:
3193:
2979:
2192:
2019:trials are “successes“ and the remaining (last)
2011:is the probability of obtaining the sequence of
1808:
16770:Journal of the American Statistical Association
16590:"Interval Estimation for a Binomial Proportion"
15489: 1. This result is a specific case of the
15456:{\displaystyle {\frac {X-np}{\sqrt {np(1-p)}}}}
15128: ≤ 8) for a binomial random variable
14715:Another commonly used rule is that both values
12255:and pulling all the terms that don't depend on
10402:{\displaystyle z=z_{\alpha /2}=z_{0.025}=-1.96}
2086:such sequences, since the binomial coefficient
16269:
16267:
8973:, which was introduced in the 18th century by
8591:{\displaystyle {\widehat {p}}={\frac {x}{n}}.}
3622:
17461:
17260:, New York: Springer-Verlag. (See especially
17222:. Cambridge University Press; First Edition.
16558:Journal of Modern Applied Statistical Methods
16426:
16383:
14617:and so, the desired first rule is satisfied,
13447:{\displaystyle {\mathcal {N}}(np,\,np(1-p)),}
11978:
11949:
11938:
11925:
11911:
11898:
11887:
11874:
7394:is a median of the binomial distribution. If
7379:
7363:
7337:
7321:
3650:is a binomially distributed random variable,
2229:
2216:
2169:
2156:
2109:
2096:
2070:
2057:
1486:
17262:Chapter X, Discrete Univariate Distributions
16633:
16583:
16581:
13310:). Conversely, any binomial distribution, B(
8789:and as the sample size approaches infinity (
7221:is a rational number (with the exception of
7164:
7146:
7083:
7080:
7062:
7042:
6987:
6978:
6966:
6957:
6868:
6847:
6835:
6808:
5796:
5778:
5655:
5633:
5554:
5548:
5528:
5507:
3116:
3110:
3022:
3016:
1684:
1578:of the number of successes in a sequence of
619:
598:
578:
557:
529:
520:
500:
491:
424:
418:
406:
400:
279:
255:
149:
125:
16741:Proceedings of the Conference CompStat 2002
16634:Agresti, Alan; Coull, Brent A. (May 1998),
16627:
16459:
16330:Advances in Mathematics: Scientific Journal
16273:
16264:
16020:(Third ed.). New York: Wiley. p.
15800:
15792:observed successes is a beta distribution.
15514:independent events each with a probability
13465:increases (at least 20) and is better when
8538:Beta distribution § Bayesian inference
8531:
8253:which implies the simpler but looser bound
6794:{\displaystyle (n+1)p-1\notin \mathbb {Z} }
5603:is equal to 0 or 1, the mode will be 0 and
3144:It can also be represented in terms of the
359:{\displaystyle {\binom {n}{k}}p^{k}q^{n-k}}
17468:
17454:
16754:
16327:
16321:
16098:
15991:. Chicago, IL, USA: Springer. p. 53.
15212:For example, suppose one randomly samples
15193:, a "proportion z-test", for the value of
15120:The following is an example of applying a
3654:being the total number of experiments and
1493:
1479:
740:{\displaystyle {\frac {q-p}{\sqrt {npq}}}}
201:– success probability for each trial
17311:A History of Mathematics: An Introduction
16605:
16578:
16569:
16495:Educational and Psychological Measurement
16341:
16194:
16136:
16104:
16034:
15341:
15201:, the sample proportion and estimator of
14675:
13619:
13416:
13281:
11190:
10573:
10549:
10498:
10147:
9701:
9677:
9652:
9521:) corresponding to the target error rate
9412:
7786:A sharper bound can be obtained from the
7614:{\displaystyle \Pr(X\geq k)=F(n-k;n,1-p)}
7543:, the probability that there are at most
6787:
5740:
5710:
5651:
3376:
2639:{\displaystyle (n+1)p-1\leq M<(n+1)p.}
1710:
17416:Binomial distribution formula calculator
17379:. New York: MacMillan. pp. 140–153.
16894:
16551:
16541:https://stats.stackexchange.com/q/275608
15986:
15795:
15282:
13347:
11346:
11212:, which is the distribution of a sum of
10686:
10409:, and one gets the upper bound by using
10203:
9596:{\displaystyle 1-{\tfrac {1}{2}}\alpha }
9503:{\displaystyle 1-{\tfrac {1}{2}}\alpha }
8891:, the posterior mean estimator becomes:
8526:
7170:{\displaystyle |m-np|\leq \min\{p,1-p\}}
2687:, with the exception of the case where (
1504:
1147:{\displaystyle g_{n}(p)={\frac {n}{pq}}}
17138:
17136:
16725:
16238:
16220:"Finding mode in Binomial distribution"
16217:
16180:
13337:
11998:the equation above can be expressed as
11143:, then the variance of the sum will be
11068:A Binomial distributed random variable
9746:
9371:Binomial proportion confidence interval
9364:
5438:is at most a constant factor away from
14:
18836:
17420:Difference of two binomial variables:
17370:
17217:
17211:
16960:
16958:
16847:(1 ed.). Springer-Verlag London.
16760:
16488:
16142:
16057:
16011:
15895:had earlier considered the case where
15140: ≤ 8) is approximated by Pr(
13932:is totally equivalent to request that
13010:. Substituting this in finally yields
7007:cannot lie too far away from the mean:
5560:{\displaystyle \lfloor \cdot \rfloor }
5534:{\displaystyle \lfloor (n+1)p\rfloor }
5470:{\displaystyle \operatorname {E} ^{c}}
3706:{\displaystyle \operatorname {E} =np.}
2383:{\displaystyle f(k,n,p)=f(n-k,n,1-p).}
631:{\displaystyle \lceil (n+1)p\rceil -1}
584:{\displaystyle \lfloor (n+1)p\rfloor }
285:{\displaystyle k\in \{0,1,\ldots ,n\}}
155:{\displaystyle n\in \{0,1,2,\ldots \}}
17449:
17411:Univariate Distribution Relationships
17258:Non-Uniform Random Variate Generation
16731:
16379:
16377:
16300:
16294:
15303:can be used as an approximation to B(
15181:independent, identically distributed
15124:. Suppose one wishes to calculate Pr(
13318:), is the distribution of the sum of
10676:) method is the most conservative. (
10088:use the Wilson (score) method below.
7536:{\displaystyle F(k;n,p)=\Pr(X\leq k)}
6170:. This proves that the mode is 0 for
3823:{\displaystyle X=X_{1}+\cdots +X_{n}}
1755:. The probability of getting exactly
18818:
17308:
17133:
17114:
16997:
16867:
16743:. Short Communications and Posters.
16303:Statistics & Probability Letters
16183:Statistics & Probability Letters
15497:
14274:, to obtain the desired conditions:
11194:{\displaystyle B(n+m,{\bar {p}}).\,}
10691:
3146:regularized incomplete beta function
2015:Bernoulli trials in which the first
1668:, not a binomial one. However, for
797:{\displaystyle {\frac {1-6pq}{npq}}}
442:regularized incomplete beta function
17357:, which must be checked separately.
16955:
16680:
16111:SIAM Journal on Applied Mathematics
13646:This can be made precise using the
11379:) (the conditional distribution of
11231:Ratio of two binomial distributions
8801:(how much depends on the priors),
5052:Stirling numbers of the second kind
3767:{\displaystyle X_{1},\ldots ,X_{n}}
2783:{\displaystyle M-p<np\leq M+1-p}
24:
17364:
17302:
17155:e-Handbook of Statistical Methods.
17069:e-Handbook of Statistical Methods.
16935:
16374:
16288:10.1111/j.1467-9574.1980.tb00681.x
16218:Nicolas, André (January 7, 2019).
15970:-ing independent Boolean variables
15948:Binomial measure, an example of a
15883:is the probability of success and
15518:of success. Mathematically, when
15399:remains fixed, the distribution of
15311:) of the binomial distribution if
13399:
13330:), each with the same probability
13150:
13056:
12846:
12773:
12581:
12514:
12389:
12321:
12248:{\displaystyle p^{k}=p^{m}p^{k-m}}
12079:
12057:
11953:
11929:
11902:
11878:
11755:
11733:
11465:then the number of balls that hit
11447:) is the number of balls that hit
11080:) can be considered as the sum of
10984:
10889:
10819:
10758:
9392:is the number of successes out of
9225:{\displaystyle {\widehat {p}}_{b}}
9154:with very rare events and a small
8812:For the special case of using the
7915:-coin (i.e. between the Bernoulli(
7099:The median is unique and equal to
5901:
5445:
5431:{\displaystyle \operatorname {E} }
5406:
5191:
5026:
4959:
4908:
4805:
4770:
4131:
4111:
3933:
3902:
3858:
3840:
3676:
3296:
3034:
2861:
2424:value can be found by calculating
2220:
2160:
2100:
2061:
1929:
1836:
506:{\displaystyle \lfloor np\rfloor }
315:
25:
18870:
17403:
17377:Introduction to Modern Statistics
16043:. New York: McGraw-Hill. p.
15987:Westland, J. Christopher (2020).
15966:, the resulting probability when
15859:This distribution was derived by
11139:do not have the same probability
10319:{\displaystyle z=z_{1-\alpha /2}}
10091:
9189:{\displaystyle {\widehat {p}}=0,}
8797:solution. The Bayes estimator is
5392:{\displaystyle c=O({\sqrt {np}})}
4082:
3122:{\displaystyle \lfloor k\rfloor }
1637:, the binomial distribution is a
1576:discrete probability distribution
1528:The probability that a ball in a
909:, use the natural log in the log.
18859:Exponential family distributions
18817:
18808:
18807:
17151:"6.3.3.1. Counts Control Charts"
16750:from the original on 2022-10-09.
16478:from the original on 2022-10-09.
16386:Bulletin of Mathematical Biology
15913:
12999:{\displaystyle (p-pq+1-p)^{n-m}}
12735:in the expression above, we get
9048:, which leads to the estimator:
7457:{\displaystyle m={\frac {n}{2}}}
7420:{\displaystyle p={\frac {1}{2}}}
7287:{\displaystyle p={\frac {1}{2}}}
7247:{\displaystyle p={\frac {1}{2}}}
7203:{\displaystyle p={\frac {1}{2}}}
4756:The non-central moments satisfy
3611:
3405:cumulative distribution function
2940:cumulative distribution function
2934:Cumulative distribution function
1199:
535:{\displaystyle \lceil np\rceil }
58:
56:Cumulative distribution function
46:
17327:
17313:. Addison-Wesley. p. 491.
17267:
17250:
17236:
17194:
17159:
17108:
17073:
17051:
17016:
16991:
16965:Box, Hunter and Hunter (1978).
16929:
16861:
16834:
16821:Engineering Statistics Handbook
16809:
16700:
16683:"Agresti-Coull Interval Method"
16674:
16545:
16529:
16482:
16453:
16420:
16251:
16232:
16210:
15863:. He considered the case where
15648:prior probability distributions
15485:with expected value 0 and
15473:with expected value 0 and
15066:
15060:
14919:
14913:
14838:are greater than 9. Since
14460:
14454:
14325:
14319:
14104:
14098:
13991:
13985:
13804:
13798:
13469:is not near to 0 or 1. Various
10279:to get the lower bound, or use
10272:{\displaystyle z=z_{\alpha /2}}
9375:Even for quite large values of
1020:{\displaystyle (q+pe^{it})^{n}}
17344:
16761:Wilson, Edwin B. (June 1927),
16435:. Dover Publications. p.
16174:
16086:
16051:
16028:
16005:
15980:
15939:Negative binomial distribution
15758:
15746:
15723:
15710:
15685:
15667:
15650:for binomial distributions in
15627:
15609:
15603:
15591:
15585:
15567:
15447:
15435:
15372:) distribution approaches the
15360:approaches 0 with the product
15259:
15247:
15189:. This fact is the basis of a
14935:
14923:
14910:
14898:
14825:
14813:
14760:
14748:
14255:
14242:
14149:
14137:
14120:
14108:
14093:
14081:
14024:
14012:
13974:
13962:
13919:
13907:
13899:
13887:
13734:
13722:
13714:
13702:
13550:
13538:
13526:
13509:
13473:may be used to decide whether
13438:
13435:
13423:
13404:
13294: = 1. Symbolically,
13263:
13248:
13200:
13184:
13175:
13165:
13118:
13090:
13081:
13071:
13039:
13027:
12981:
12953:
12907:
12894:
12885:
12869:
12798:
12788:
12760:
12748:
12666:
12653:
12633:
12621:
12539:
12529:
12477:
12464:
12449:
12436:
12304:
12292:
12171:
12158:
12143:
12130:
12023:
12011:
11831:
11818:
11803:
11790:
11692:
11680:
11674:
11650:
11616:
11604:
11570:
11558:
11529:
11517:
11406:For example, imagine throwing
11184:
11178:
11157:
11108:) is equivalent to the sum of
11030:
11017:
10941:
10928:
10857:
10844:
10776:
10764:
10581:
10557:
9879:
9821:
9815:
9800:
9794:
9772:
9709:
9685:
9627:
9450:is the proportion of successes
9035:
8997:
8878:
8866:
8857:
8833:
8688:
8676:
8601:This estimator is found using
8428:
8401:
8284:
8266:
8180:
8159:
8129:
8111:
8081:
8063:
7994:
7982:
7960:
7954:
7948:
7936:
7821:
7803:
7666:
7648:
7608:
7578:
7569:
7557:
7530:
7518:
7509:
7491:
7468:
7136:
7119:
7032:
7015:
6862:
6850:
6823:
6811:
6771:
6759:
6736:
6724:
6695:
6683:
6654:
6642:
6615:
6609:
6600:
6588:
6582:
6570:
6558:
6545:
6539:
6530:
6518:
6512:
6500:
6488:
6475:
6469:
6460:
6448:
6442:
6430:
6418:
6378:
6366:
6363:
6351:
6343:
6331:
6319:
6313:
6305:
6293:
6125:
6119:
6090:
6084:
6029:
6023:
6000:
5994:
5888:
5882:
5828:
5816:
5769:
5757:
5737:
5725:
5707:
5695:
5677:
5665:
5648:
5636:
5522:
5510:
5458:
5451:
5425:
5412:
5386:
5373:
5291:
5281:
5262:
5253:
5244:
5233:
5210:
5197:
5118:
5100:
5094:
5082:
4927:
4914:
4852:
4840:
4824:
4811:
4782:
4776:
4733:
4724:
4711:
4682:
4679:
4667:
4652:
4649:
4637:
4622:
4619:
4607:
4592:
4589:
4577:
4562:
4559:
4547:
4514:
4511:
4499:
4493:
4478:
4469:
4466:
4451:
4448:
4436:
4403:
4400:
4388:
4382:
4367:
4358:
4355:
4343:
4310:
4295:
4292:
4280:
4247:
4235:
4147:
4143:
4137:
4122:
4060:
4048:
4024:
4018:
3952:
3939:
3921:
3908:
3896:
3864:
3852:
3846:
3688:
3682:
3586:
3574:
3552:
3540:
3445:
3427:
3367:
3354:
3286:
3274:
3261:
3237:
3208:
3196:
3183:
3165:
3072:
3059:
2994:
2982:
2973:
2955:
2899:
2886:
2848:
2830:
2627:
2615:
2594:
2582:
2549:
2537:
2534:
2522:
2514:
2502:
2490:
2472:
2464:
2440:
2420:value that maximizes it. This
2374:
2344:
2335:
2317:
2258:
2245:
2207:
2195:
1976:
1964:
1874:
1861:
1823:
1811:
1802:
1784:
1679:
1266:Collectively exhaustive events
1123:
1117:
1073:
1057:
1051:
1045:
1008:
982:
960:{\displaystyle (q+pe^{t})^{n}}
948:
925:
863:
842:
683:
671:
613:
601:
572:
560:
427:
391:
188:
176:
91:
79:
32:Binomial options pricing model
13:
1:
18854:Conjugate prior distributions
18849:Factorial and binomial topics
16161:10.1080/00031305.2019.1679257
15974:
15849:pseudorandom number generator
15807:Pseudo-random number sampling
15147:This approximation, known as
13269:{\displaystyle Y\sim B(n,pq)}
11210:Poisson binomial distribution
11204:Poisson binomial distribution
10667:
10016:{\displaystyle n_{1}\neq 0,n}
8814:standard uniform distribution
6950:must lie within the interval
3617:
16969:Statistics for experimenters
16883:(2): 295–312. Archived from
16315:10.1016/0167-7152(94)00090-U
14051:Moving terms around yields:
13358:probability density function
11576:{\displaystyle Y\sim B(X,q)}
11535:{\displaystyle X\sim B(n,p)}
9515:standard normal distribution
9396:, the total number of trials
8603:maximum likelihood estimator
6752:is a mode. In the case that
3403:which is equivalent to the
2677:and monotone decreasing for
2293:values. This is because for
2178:{\textstyle {\binom {n}{k}}}
2118:{\textstyle {\binom {n}{k}}}
2079:{\textstyle {\binom {n}{k}}}
1717:{\displaystyle \mathbb {N} }
1630:; for a single trial, i.e.,
1539:ends up in the central bin (
7:
16105:Knoblauch, Andreas (2008),
15906:
14863:{\displaystyle 0<p<1}
14195:{\displaystyle 0<p<1}
9958:This method works well for
8652:also exists when using the
8637:both in probability and in
6270:{\displaystyle 0<p<1}
5500:) distribution is equal to
3623:Expected value and variance
2667:is monotone increasing for
1666:hypergeometric distribution
689:{\displaystyle npq=np(1-p)}
292:– number of successes
10:
18875:
18641:Wrapped asymmetric Laplace
17612:Extended negative binomial
17371:Hirsch, Werner Z. (1957).
16507:10.1177/001316447903900302
16462:"The Probabilistic Method"
16460:Matoušek, J.; Vondrak, J.
16024:(theorem in section VI.3).
15944:Beta-binomial distribution
15854:
15804:
15315:is sufficiently large and
14267:{\displaystyle n(1-p)^{2}}
13341:
13302:) has the same meaning as
12275:out of the sum now yields
11458:is the probability to hit
11428:is the probability to hit
11216:independent non-identical
10207:
10095:
9750:
9631:
9368:
8535:
7258:odd) the median is unique.
7177:(except for the case when
5685: is 0 or a noninteger
2808:
2393:Looking at the expression
1660:from a population of size
1509:Binomial distribution for
29:
18803:
18737:
18695:
18596:
18432:
18410:
18401:
18300:Generalized extreme value
18285:
18120:
18080:Relativistic Breit–Wigner
17796:
17693:
17684:
17577:
17497:
17488:
17477:Probability distributions
17276:Communications of the ACM
16643:The American Statistician
16571:10.22237/jmasm/1036110000
16205:10.1016/j.spl.2021.109306
16149:The American Statistician
16035:Wadsworth, G. P. (1960).
15646:also provide a family of
15388:de Moivre–Laplace theorem
15364:held fixed, the Binomial(
15149:de Moivre–Laplace theorem
13354:probability mass function
10081:{\displaystyle n_{1}=0,n}
9541:. For example, for a 95%
8621:, since it is based on a
6884:
6013:has a nonzero value with
1769:probability mass function
1685:Probability mass function
1100:
1095:
1085:{\displaystyle G(z)=^{n}}
1035:
1030:
975:
970:
918:
913:
812:
807:
755:
750:
704:
699:
646:
641:
550:
545:
484:
479:
453:
448:
374:
369:
301:
296:
242:
237:
112:
107:
69:
66:
54:
44:Probability mass function
42:
17350:Except the trivial case
16552:Razzaghi, Mehdi (2002).
16489:Wilcox, Rand R. (1979).
16259:The Mathematical Gazette
15934:Multinomial distribution
15813:random number generation
15801:Random number generation
15788:independent events with
14780:Assume that both values
11585:law of total probability
10042:{\displaystyle n\leq 10}
8793:→ ∞), it approaches the
8787:asymptotically efficient
8546:is known, the parameter
8532:Estimation of parameters
8087:{\displaystyle F(k;n,p)}
7635:yields the simple bound
6710:{\displaystyle (n+1)p-1}
6669:{\displaystyle (n+1)p-1}
1647:statistical significance
1592:, and each with its own
1436:Law of total probability
1431:Conditional independence
1320:Exponential distribution
1305:Probability distribution
162:– number of trials
27:Probability distribution
18295:Generalized chi-squared
18239:Normal-inverse Gaussian
17117:H2 mathematics handbook
17000:H2 Mathematics Handbook
15891:are positive integers.
15162:The Doctrine of Chances
14870:, we easily have that
13306: ~ Bernoulli(
13298: ~ B(1,
10672:The so-called "exact" (
10339:{\displaystyle \alpha }
9977:{\displaystyle n>10}
9603: = 0.975 and
9558:{\displaystyle \alpha }
9534:{\displaystyle \alpha }
8785:The Bayes estimator is
8663:. When using a general
8619:Lehmann–Scheffé theorem
6163:{\displaystyle k\neq n}
5479:
1415:Conditional probability
18844:Discrete distributions
18607:Univariate (circular)
18168:Generalized hyperbolic
17597:Conway–Maxwell–Poisson
17587:Beta negative binomial
17218:MacKay, David (2003).
16817:"Confidence intervals"
16708:"Confidence intervals"
16687:pellucid.atlassian.net
16427:Robert B. Ash (1990).
16276:Statistica Neerlandica
15771:
15634:
15457:
15342:Limiting distributions
15273:
15109:
15009:
14983:
14960:
14864:
14832:
14831:{\displaystyle n(1-p)}
14797:
14767:
14766:{\displaystyle n(1-p)}
14732:
14703:
14608:
14506:
14397:
14396:{\displaystyle n>9}
14368:
14268:
14226:
14225:{\displaystyle np^{2}}
14196:
14161:
14042:
13926:
13847:
13744:
13635:
13448:
13369:
13288:Bernoulli distribution
13282:Bernoulli distribution
13270:
13223:
13000:
12937:
12838:
12729:
12694:
12573:
12381:
12269:
12249:
12190:
12049:
11992:
11854:
11725:
11646:
11577:
11536:
11195:
11059:
10806:
10656:
10469:
10403:
10340:
10320:
10273:
10194:
10082:
10043:
10017:
9978:
9949:
9850:
9726:
9617:
9597:
9565: = 0.05, so
9559:
9535:
9504:
9466:
9444:
9355:
9289:
9226:
9190:
9141:
9042:
8956:
8885:
8776:
8695:
8592:
8517:
8363:
8244:
8088:
8037:
7882:
7738:
7633:Hoeffding's inequality
7615:
7537:
7458:
7421:
7388:
7288:
7248:
7204:
7171:
7090:
6994:
6937:
6914:
6875:
6795:
6746:
6745:{\displaystyle (n+1)p}
6711:
6670:
6626:
6388:
6271:
6236:
6210:
6190:
6164:
6138:
6137:{\displaystyle f(k)=0}
6103:
6102:{\displaystyle f(n)=1}
6068:
6042:
6041:{\displaystyle f(0)=1}
6007:
5978:
5949:
5857:
5561:
5535:
5471:
5432:
5393:
5346:
5177:higher Poisson moments
5169:
5145:
5125:
5044:
5004:
4953:
4889:
4747:
4165:
4070:
3989:
3824:
3768:
3728:. In other words, if
3707:
3601:
3394:
3137:less than or equal to
3123:
3094:
3026:
2924:
2784:
2640:
2559:
2384:
2277:
2179:
2119:
2080:
1989:
1893:
1718:
1639:Bernoulli distribution
1551:
1357:Continuous or discrete
1310:Bernoulli distribution
1169:
1148:
1086:
1021:
961:
894:
798:
741:
690:
632:
585:
536:
507:
470:
434:
360:
286:
228:
195:
156:
98:
97:{\displaystyle B(n,p)}
18652:Bivariate (spherical)
18150:Kaniadakis κ-Gaussian
17409:Interactive graphic:
16823:. NIST/Sematech. 2012
16732:Pires, M. A. (2002).
16616:10.1214/ss/1009213286
15959:Statistical mechanics
15817:marginal distribution
15805:Further information:
15796:Computational methods
15772:
15635:
15491:central limit theorem
15483:asymptotically normal
15458:
15349:Poisson limit theorem
15331: ≥ 100 and
15283:Poisson approximation
15274:
15207:common test statistic
15167:central limit theorem
15122:continuity correction
15110:
15010:
14984:
14961:
14865:
14833:
14798:
14768:
14733:
14704:
14609:
14507:
14398:
14369:
14269:
14227:
14197:
14162:
14043:
13927:
13848:
13745:
13636:
13485:One rule is that for
13477:is large enough, and
13459:continuity correction
13449:
13351:
13271:
13224:
13001:
12938:
12812:
12730:
12728:{\displaystyle i=k-m}
12695:
12553:
12361:
12270:
12250:
12191:
12029:
11993:
11855:
11705:
11626:
11578:
11537:
11347:Conditional binomials
11196:
11060:
10786:
10687:Related distributions
10657:
10470:
10404:
10341:
10321:
10274:
10204:Wilson (score) method
10195:
10083:
10044:
10018:
9979:
9950:
9859:Here the estimate of
9851:
9737:continuity correction
9727:
9618:
9598:
9560:
9536:
9505:
9467:
9445:
9356:
9290:
9227:
9191:
9142:
9043:
8957:
8886:
8818:non-informative prior
8777:
8696:
8593:
8527:Statistical inference
8518:
8364:
8245:
8098:it can be shown that
8089:
8038:
7883:
7739:
7616:
7538:
7464:is the unique median.
7459:
7422:
7389:
7289:
7249:
7205:
7172:
7091:
6995:
6938:
6915:
6876:
6796:
6747:
6712:
6671:
6627:
6389:
6272:
6237:
6211:
6191:
6165:
6139:
6104:
6069:
6043:
6008:
5979:
5950:
5858:
5599: − 1. When
5562:
5536:
5472:
5433:
5394:
5347:
5170:
5146:
5126:
5045:
5005:
4933:
4890:
4748:
4166:
4071:
3990:
3825:
3769:
3708:
3602:
3395:
3129:is the "floor" under
3124:
3095:
3000:
2942:can be expressed as:
2925:
2785:
2641:
2560:
2385:
2278:
2180:
2120:
2081:
1990:
1894:
1719:
1564:binomial distribution
1508:
1315:Binomial distribution
1170:
1149:
1087:
1022:
962:
895:
799:
742:
691:
633:
586:
537:
508:
471:
435:
361:
287:
229:
227:{\displaystyle q=1-p}
196:
194:{\displaystyle p\in }
157:
99:
38:Binomial distribution
18717:Dirac delta function
18664:Bivariate (toroidal)
18621:Univariate von Mises
18492:Multivariate Laplace
18384:Shifted log-logistic
17733:Continuous Bernoulli
17256:Devroye, Luc (1986)
16868:Wang, Y. H. (1993).
16352:10.37418/amsj.10.4.9
16143:Nguyen, Duy (2021),
15661:
15558:
15409:
15374:Poisson distribution
15323: ≥ 20 and
15289:Poisson distribution
15231:
15185:with parameter
15022:
14993:
14973:
14877:
14842:
14807:
14784:
14742:
14719:
14624:
14522:
14410:
14381:
14281:
14236:
14206:
14174:
14058:
13939:
13864:
13760:
13664:
13648:Berry–Esseen theorem
13502:
13394:
13338:Normal approximation
13236:
13017:
12950:
12742:
12707:
12282:
12259:
12203:
12005:
11867:
11594:
11546:
11505:
11274:be independent. Let
11151:
10751:
10485:
10413:
10350:
10330:
10283:
10242:
10108:
10053:
10027:
9988:
9962:
9870:
9763:
9747:Agresti–Coull method
9642:
9607:
9569:
9549:
9525:
9476:
9456:
9402:
9365:Confidence intervals
9313:
9239:
9200:
9162:
9055:
8988:
8975:Pierre-Simon Laplace
8898:
8824:
8712:
8667:
8609:. This estimator is
8557:
8395:
8260:
8105:
8057:
8049:One can also obtain
7930:
7797:
7642:
7551:
7485:
7435:
7398:
7306:
7265:
7225:
7181:
7115:
7011:
6954:
6924:
6901:
6805:
6756:
6721:
6680:
6676:is an integer, then
6639:
6405:
6284:
6249:
6220:
6200:
6174:
6148:
6113:
6078:
6052:
6017:
6006:{\displaystyle f(0)}
5988:
5962:
5876:
5614:
5545:
5504:
5442:
5403:
5361:
5188:
5159:
5135:
5058:
5017:
4905:
4763:
4178:
4095:
4009:
3837:
3782:
3732:
3673:
3421:
3155:
3107:
2949:
2824:
2741:
2579:
2431:
2311:
2189:
2150:
2129:successes among the
2090:
2051:
2000:binomial coefficient
1919:
1778:
1706:
1441:Law of large numbers
1410:Marginal probability
1335:Poisson distribution
1184:Part of a series on
1159:
1104:
1039:
979:
922:
816:
759:
708:
650:
595:
554:
517:
488:
457:
378:
305:
246:
206:
167:
116:
73:
18765:Natural exponential
18670:Bivariate von Mises
18636:Wrapped exponential
18502:Multivariate stable
18497:Multivariate normal
17818:Benktander 2nd kind
17813:Benktander 1st kind
17602:Discrete phase-type
17288:10.1145/42372.42381
17244:"Beta distribution"
16594:Statistical Science
16012:Feller, W. (1968).
15929:Logistic regression
15471:normal distribution
15395:approaches ∞ while
15183:Bernoulli variables
15008:{\displaystyle 1-p}
13386:normal distribution
13364: = 6 and
12703:After substituting
9302:obtained using the
9300:confidence interval
8613:and uniformly with
8053:bounds on the tail
7298:is odd, any number
6235:{\displaystyle p=1}
6189:{\displaystyle p=0}
6067:{\displaystyle p=1}
5977:{\displaystyle p=0}
5357:This shows that if
3331:
1689:In general, if the
1400:Complementary event
1342:Probability measure
1330:Pareto distribution
1325:Normal distribution
39:
18420:Rectified Gaussian
18305:Generalized Pareto
18163:Generalized normal
18035:Matrix-exponential
17386:Applied Statistics
17115:Chen, Zac (2011).
16998:Chen, Zac (2011).
16431:Information Theory
16398:10.1007/BF02458840
15921:Mathematics portal
15767:
15652:Bayesian inference
15644:Beta distributions
15630:
15453:
15269:
15105:
15005:
14979:
14956:
14860:
14828:
14796:{\displaystyle np}
14793:
14763:
14731:{\displaystyle np}
14728:
14699:
14604:
14502:
14393:
14364:
14264:
14222:
14192:
14157:
14038:
13922:
13843:
13740:
13631:
13444:
13384:) is given by the
13370:
13360:approximation for
13266:
13219:
13217:
12996:
12933:
12725:
12690:
12688:
12265:
12245:
12186:
11988:
11983:
11943:
11916:
11892:
11850:
11848:
11573:
11532:
11410:balls to a basket
11191:
11055:
11053:
10652:
10465:
10399:
10336:
10316:
10269:
10190:
10078:
10039:
10013:
9974:
9945:
9846:
9722:
9623: = 1.96.
9613:
9593:
9588:
9555:
9531:
9500:
9495:
9462:
9440:
9351:
9285:
9222:
9186:
9137:
9038:
8971:rule of succession
8952:
8881:
8772:
8691:
8661:prior distribution
8623:minimal sufficient
8588:
8513:
8426:
8359:
8240:
8178:
8157:
8096:Stirling's formula
8084:
8033:
7878:
7751:= 1, we have that
7734:
7611:
7533:
7454:
7417:
7384:
7284:
7244:
7200:
7167:
7086:
6990:
6936:{\displaystyle np}
6933:
6913:{\displaystyle np}
6910:
6871:
6791:
6742:
6707:
6666:
6622:
6620:
6398:From this follows
6384:
6267:
6232:
6206:
6186:
6160:
6134:
6099:
6064:
6038:
6003:
5974:
5945:
5853:
5848:
5579:is an integer and
5557:
5531:
5467:
5428:
5389:
5342:
5165:
5141:
5121:
5073:
5040:
5039:
5000:
4984:
4885:
4883:
4743:
4741:
4161:
4066:
3985:
3820:
3764:
3703:
3597:
3390:
3388:
3311:
3119:
3090:
2920:
2780:
2636:
2555:
2380:
2273:
2175:
2115:
2076:
1985:
1889:
1767:) is given by the
1714:
1612:(with probability
1604:(with probability
1556:probability theory
1552:
1451:Boole's inequality
1387:Stochastic process
1276:Mutual exclusivity
1193:Probability theory
1165:
1144:
1097:Fisher information
1082:
1017:
957:
890:
794:
737:
686:
628:
581:
532:
503:
469:{\displaystyle np}
466:
430:
356:
282:
224:
191:
152:
94:
37:
18831:
18830:
18428:
18427:
18397:
18396:
18288:whose type varies
18234:Normal (Gaussian)
18188:Hyperbolic secant
18137:Exponential power
18040:Maxwell–Boltzmann
17788:Wigner semicircle
17680:
17679:
17652:Parabolic fractal
17642:Negative binomial
17441:causaScientia.org
17437:online calculator
17320:978-0-321-38700-4
16973:. Wiley. p.
16877:Statistica Sinica
16854:978-1-84628-168-6
16681:Gulotta, Joseph.
16123:10.1137/070700024
15998:978-3-030-49091-1
15901:Pascal's triangle
15762:
15498:Beta distribution
15451:
15450:
15356:approaches ∞ and
15335: ≤ 10.
15267:
15266:
15157:Abraham de Moivre
15096:
15064:
15054:
14982:{\displaystyle p}
14917:
14694:
14690:
14673:
14672:
14650:
14649:
14599:
14595:
14581:
14580:
14558:
14557:
14535:
14531:
14494:
14493:
14471:
14467:
14458:
14446:
14445:
14423:
14419:
14355:
14323:
14313:
14152:
14102:
14096:
14027:
13989:
13977:
13902:
13834:
13802:
13792:
13717:
13617:
13616:
13594:
13593:
13569:
13568:
13554:
13553:
13157:
13063:
12861:
12780:
12604:
12521:
12412:
12328:
12268:{\displaystyle k}
12102:
12064:
11976:
11936:
11909:
11885:
11762:
11740:
11181:
10999:
10904:
10826:
10692:Sums of binomials
10650:
10647:
10622:
10620:
10588:
10578:
10554:
10529:
10503:
10180:
10177:
10154:
10152:
9943:
9912:
9882:
9844:
9843:
9818:
9797:
9775:
9717:
9716:
9706:
9682:
9657:
9616:{\displaystyle z}
9587:
9494:
9465:{\displaystyle z}
9438:
9417:
9346:
9332:
9326:
9280:
9252:
9213:
9174:
9132:
9118:
9068:
9033:
9014:
8947:
8911:
8767:
8725:
8654:Beta distribution
8629:statistic (i.e.:
8607:method of moments
8583:
8569:
8491:
8478:
8442:
8425:
8338:
8303:
8302:
8219:
8184:
8183:
8177:
8156:
8027:
7977:
7923:) distribution):
7860:
7767:) = 0 (for fixed
7712:
7547:successes. Since
7452:
7415:
7359:
7317:
7282:
7242:
7198:
6382:
6323:
6209:{\displaystyle n}
5908:
5814:
5755:
5724:
5720:
5716:
5686:
5663:
5620:
5571:. However, when (
5384:
5333:
5266:
5168:{\displaystyle n}
5144:{\displaystyle k}
5066:
5033:
4977:
4966:
3519:
3493:
3462:
3303:
3041:
2868:
2553:
2494:
2412:as a function of
2227:
2167:
2107:
2068:
1983:
1936:
1843:
1672:much larger than
1628:Bernoulli process
1524:Pascal's triangle
1503:
1502:
1405:Joint probability
1352:Bernoulli process
1251:Probability space
1179:
1178:
1168:{\displaystyle n}
1142:
884:
827:
792:
735:
734:
322:
16:(Redirected from
18866:
18821:
18820:
18811:
18810:
18750:Compound Poisson
18725:
18713:
18682:von Mises–Fisher
18678:
18666:
18654:
18616:Circular uniform
18612:
18532:
18476:
18447:
18408:
18407:
18310:Marchenko–Pastur
18173:Geometric stable
18090:Truncated normal
17983:Inverse Gaussian
17889:Hyperexponential
17728:Beta rectangular
17696:bounded interval
17691:
17690:
17559:Discrete uniform
17544:Poisson binomial
17495:
17494:
17470:
17463:
17456:
17447:
17446:
17399:
17380:
17358:
17356:
17348:
17337:
17331:
17325:
17324:
17306:
17300:
17299:
17271:
17265:
17254:
17248:
17247:
17240:
17234:
17233:
17215:
17209:
17198:
17192:
17191:
17185:
17177:
17175:
17174:
17163:
17157:
17140:
17131:
17130:
17112:
17106:
17105:
17099:
17091:
17089:
17088:
17077:
17071:
17055:
17049:
17048:
17042:
17034:
17032:
17031:
17020:
17014:
17013:
16995:
16989:
16988:
16972:
16962:
16953:
16952:
16950:
16948:
16933:
16927:
16926:
16898:
16892:
16891:
16889:
16874:
16865:
16859:
16858:
16838:
16832:
16831:
16829:
16828:
16813:
16807:
16806:
16805:
16804:
16798:
16792:, archived from
16776:(158): 209–212,
16767:
16758:
16752:
16751:
16749:
16738:
16729:
16723:
16722:
16720:
16718:
16704:
16698:
16697:
16695:
16693:
16678:
16672:
16671:
16670:
16669:
16640:
16631:
16625:
16624:
16623:
16622:
16609:
16585:
16576:
16575:
16573:
16549:
16543:
16533:
16527:
16526:
16486:
16480:
16479:
16477:
16466:
16457:
16451:
16450:
16434:
16424:
16418:
16417:
16381:
16372:
16371:
16345:
16336:(4): 1951–1958.
16325:
16319:
16318:
16298:
16292:
16291:
16271:
16262:
16255:
16249:
16248:
16236:
16230:
16229:
16214:
16208:
16207:
16198:
16178:
16172:
16171:
16140:
16134:
16133:
16102:
16096:
16090:
16084:
16083:
16055:
16049:
16048:
16042:
16032:
16026:
16025:
16019:
16009:
16003:
16002:
15984:
15923:
15918:
15917:
15846:
15842:
15838:
15834:
15791:
15787:
15783:
15776:
15774:
15773:
15768:
15763:
15761:
15738:
15737:
15736:
15709:
15708:
15692:
15639:
15637:
15636:
15631:
15550:
15543:
15528:
15517:
15513:
15510:successes given
15509:
15462:
15460:
15459:
15454:
15452:
15428:
15427:
15413:
15278:
15276:
15275:
15270:
15268:
15262:
15242:
15241:
15114:
15112:
15111:
15106:
15101:
15097:
15095:
15081:
15065:
15062:
15059:
15055:
15050:
15039:
15014:
15012:
15011:
15006:
14988:
14986:
14985:
14980:
14965:
14963:
14962:
14957:
14918:
14915:
14869:
14867:
14866:
14861:
14837:
14835:
14834:
14829:
14802:
14800:
14799:
14794:
14772:
14770:
14769:
14764:
14737:
14735:
14734:
14729:
14708:
14706:
14705:
14700:
14695:
14686:
14685:
14680:
14676:
14674:
14671:
14657:
14656:
14651:
14645:
14634:
14633:
14613:
14611:
14610:
14605:
14600:
14591:
14590:
14582:
14579:
14565:
14564:
14559:
14553:
14542:
14541:
14536:
14527:
14526:
14511:
14509:
14508:
14503:
14495:
14492:
14478:
14477:
14472:
14463:
14462:
14459:
14456:
14447:
14441:
14430:
14429:
14424:
14415:
14414:
14402:
14400:
14399:
14394:
14373:
14371:
14370:
14365:
14360:
14356:
14354:
14340:
14324:
14321:
14318:
14314:
14309:
14298:
14273:
14271:
14270:
14265:
14263:
14262:
14231:
14229:
14228:
14223:
14221:
14220:
14201:
14199:
14198:
14193:
14166:
14164:
14163:
14158:
14153:
14130:
14103:
14100:
14097:
14074:
14047:
14045:
14044:
14039:
14028:
14005:
13990:
13987:
13978:
13955:
13931:
13929:
13928:
13923:
13903:
13880:
13852:
13850:
13849:
13844:
13839:
13835:
13833:
13819:
13803:
13800:
13797:
13793:
13788:
13777:
13749:
13747:
13746:
13741:
13718:
13695:
13640:
13638:
13637:
13632:
13624:
13620:
13618:
13615:
13601:
13600:
13595:
13589:
13578:
13577:
13570:
13564:
13560:
13555:
13531:
13530:
13529:
13512:
13506:
13491:
13453:
13451:
13450:
13445:
13403:
13402:
13368: = 0.5
13324:Bernoulli trials
13275:
13273:
13272:
13267:
13228:
13226:
13225:
13220:
13218:
13214:
13213:
13183:
13182:
13164:
13163:
13162:
13149:
13136:
13132:
13131:
13089:
13088:
13070:
13069:
13068:
13055:
13008:binomial theorem
13005:
13003:
13002:
12997:
12995:
12994:
12942:
12940:
12939:
12934:
12932:
12928:
12927:
12926:
12893:
12892:
12868:
12867:
12866:
12857:
12845:
12837:
12826:
12806:
12805:
12787:
12786:
12785:
12772:
12734:
12732:
12731:
12726:
12699:
12697:
12696:
12691:
12689:
12685:
12681:
12680:
12679:
12652:
12651:
12640:
12636:
12611:
12610:
12609:
12603:
12592:
12580:
12572:
12567:
12547:
12546:
12528:
12527:
12526:
12513:
12500:
12496:
12492:
12491:
12490:
12463:
12462:
12435:
12434:
12419:
12418:
12417:
12411:
12400:
12388:
12380:
12375:
12355:
12354:
12345:
12344:
12335:
12334:
12333:
12320:
12274:
12272:
12271:
12266:
12254:
12252:
12251:
12246:
12244:
12243:
12228:
12227:
12215:
12214:
12195:
12193:
12192:
12187:
12185:
12184:
12157:
12156:
12129:
12128:
12119:
12118:
12109:
12108:
12107:
12101:
12090:
12078:
12071:
12070:
12069:
12056:
12048:
12043:
11997:
11995:
11994:
11989:
11984:
11982:
11981:
11975:
11964:
11952:
11944:
11942:
11941:
11928:
11917:
11915:
11914:
11901:
11893:
11891:
11890:
11877:
11859:
11857:
11856:
11851:
11849:
11845:
11844:
11817:
11816:
11789:
11788:
11779:
11778:
11769:
11768:
11767:
11754:
11747:
11746:
11745:
11732:
11724:
11719:
11698:
11645:
11640:
11582:
11580:
11579:
11574:
11541:
11539:
11538:
11533:
11484:) and therefore
11342:
11296:
11273:
11255:
11218:Bernoulli trials
11200:
11198:
11197:
11192:
11183:
11182:
11174:
11064:
11062:
11061:
11056:
11054:
11050:
11049:
11016:
11015:
11006:
11005:
11004:
10995:
10983:
10970:
10966:
10962:
10961:
10960:
10927:
10926:
10911:
10910:
10909:
10903:
10888:
10876:
10872:
10871:
10870:
10843:
10842:
10833:
10832:
10831:
10818:
10805:
10800:
10661:
10659:
10658:
10653:
10651:
10649:
10648:
10643:
10642:
10633:
10624:
10623:
10621:
10619:
10618:
10617:
10604:
10603:
10594:
10589:
10584:
10580:
10579:
10574:
10568:
10556:
10555:
10550:
10544:
10540:
10538:
10530:
10528:
10520:
10519:
10510:
10505:
10504:
10499:
10493:
10489:
10474:
10472:
10471:
10466:
10458:
10457:
10445:
10444:
10440:
10408:
10406:
10405:
10400:
10389:
10388:
10376:
10375:
10371:
10345:
10343:
10342:
10337:
10325:
10323:
10322:
10317:
10315:
10314:
10310:
10278:
10276:
10275:
10270:
10268:
10267:
10263:
10199:
10197:
10196:
10191:
10186:
10182:
10181:
10179:
10178:
10173:
10164:
10159:
10155:
10153:
10148:
10142:
10140:
10120:
10119:
10087:
10085:
10084:
10079:
10065:
10064:
10048:
10046:
10045:
10040:
10022:
10020:
10019:
10014:
10000:
9999:
9983:
9981:
9980:
9975:
9954:
9952:
9951:
9946:
9944:
9942:
9941:
9940:
9924:
9923:
9922:
9913:
9905:
9900:
9899:
9889:
9884:
9883:
9875:
9855:
9853:
9852:
9847:
9845:
9842:
9841:
9840:
9824:
9820:
9819:
9811:
9799:
9798:
9790:
9786:
9785:
9777:
9776:
9768:
9731:
9729:
9728:
9723:
9718:
9712:
9708:
9707:
9702:
9696:
9684:
9683:
9678:
9672:
9668:
9667:
9659:
9658:
9653:
9647:
9622:
9620:
9619:
9614:
9602:
9600:
9599:
9594:
9589:
9580:
9564:
9562:
9561:
9556:
9543:confidence level
9540:
9538:
9537:
9532:
9509:
9507:
9506:
9501:
9496:
9487:
9471:
9469:
9468:
9463:
9449:
9447:
9446:
9441:
9439:
9434:
9433:
9424:
9419:
9418:
9413:
9407:
9360:
9358:
9357:
9352:
9347:
9339:
9334:
9333:
9330:
9328:
9327:
9319:
9294:
9292:
9291:
9286:
9281:
9279:
9265:
9260:
9259:
9254:
9253:
9245:
9231:
9229:
9228:
9223:
9221:
9220:
9215:
9214:
9206:
9195:
9193:
9192:
9187:
9176:
9175:
9167:
9150:When estimating
9146:
9144:
9143:
9138:
9133:
9131:
9120:
9119:
9111:
9102:
9097:
9096:
9070:
9069:
9061:
9047:
9045:
9044:
9039:
9034:
9026:
9015:
9007:
8980:When relying on
8961:
8959:
8958:
8953:
8948:
8946:
8935:
8924:
8919:
8918:
8913:
8912:
8904:
8890:
8888:
8887:
8882:
8809:in probability.
8781:
8779:
8778:
8773:
8768:
8766:
8749:
8738:
8733:
8732:
8727:
8726:
8718:
8701:as a prior, the
8700:
8698:
8697:
8692:
8615:minimum variance
8597:
8595:
8594:
8589:
8584:
8576:
8571:
8570:
8562:
8522:
8520:
8519:
8514:
8508:
8504:
8503:
8502:
8497:
8493:
8492:
8484:
8479:
8471:
8443:
8435:
8427:
8418:
8368:
8366:
8365:
8360:
8355:
8351:
8350:
8346:
8339:
8331:
8304:
8295:
8291:
8249:
8247:
8246:
8241:
8236:
8232:
8231:
8227:
8220:
8212:
8185:
8179:
8170:
8158:
8149:
8140:
8136:
8093:
8091:
8090:
8085:
8042:
8040:
8039:
8034:
8028:
8026:
8015:
8004:
7978:
7970:
7919:) and Bernoulli(
7887:
7885:
7884:
7879:
7877:
7873:
7872:
7868:
7861:
7853:
7779: <
7743:
7741:
7740:
7735:
7729:
7725:
7724:
7723:
7718:
7714:
7713:
7705:
7620:
7618:
7617:
7612:
7542:
7540:
7539:
7534:
7463:
7461:
7460:
7455:
7453:
7445:
7426:
7424:
7423:
7418:
7416:
7408:
7393:
7391:
7390:
7385:
7383:
7382:
7367:
7366:
7360:
7352:
7341:
7340:
7325:
7324:
7318:
7310:
7302:in the interval
7293:
7291:
7290:
7285:
7283:
7275:
7253:
7251:
7250:
7245:
7243:
7235:
7209:
7207:
7206:
7201:
7199:
7191:
7176:
7174:
7173:
7168:
7139:
7122:
7095:
7093:
7092:
7087:
7055:
7035:
7018:
6999:
6997:
6996:
6991:
6942:
6940:
6939:
6934:
6919:
6917:
6916:
6911:
6880:
6878:
6877:
6872:
6800:
6798:
6797:
6792:
6790:
6751:
6749:
6748:
6743:
6716:
6714:
6713:
6708:
6675:
6673:
6672:
6667:
6631:
6629:
6628:
6623:
6621:
6393:
6391:
6390:
6385:
6383:
6381:
6349:
6329:
6324:
6322:
6308:
6288:
6276:
6274:
6273:
6268:
6241:
6239:
6238:
6233:
6215:
6213:
6212:
6207:
6195:
6193:
6192:
6187:
6169:
6167:
6166:
6161:
6143:
6141:
6140:
6135:
6108:
6106:
6105:
6100:
6073:
6071:
6070:
6065:
6047:
6045:
6044:
6039:
6012:
6010:
6009:
6004:
5983:
5981:
5980:
5975:
5954:
5952:
5951:
5946:
5941:
5940:
5925:
5924:
5915:
5914:
5913:
5900:
5862:
5860:
5859:
5854:
5852:
5851:
5815:
5812:
5756:
5753:
5722:
5721:
5718:
5714:
5687:
5684:
5664:
5661:
5621:
5618:
5566:
5564:
5563:
5558:
5540:
5538:
5537:
5532:
5476:
5474:
5473:
5468:
5466:
5465:
5437:
5435:
5434:
5429:
5424:
5423:
5398:
5396:
5395:
5390:
5385:
5377:
5351:
5349:
5348:
5343:
5338:
5334:
5332:
5321:
5320:
5311:
5299:
5298:
5277:
5276:
5271:
5267:
5265:
5243:
5222:
5209:
5208:
5174:
5172:
5171:
5166:
5150:
5148:
5147:
5142:
5130:
5128:
5127:
5122:
5075:
5074:
5049:
5047:
5046:
5041:
5038:
5034:
5009:
5007:
5006:
5001:
4996:
4995:
4986:
4985:
4971:
4967:
4952:
4947:
4926:
4925:
4894:
4892:
4891:
4886:
4884:
4877:
4876:
4867:
4866:
4823:
4822:
4752:
4750:
4749:
4744:
4742:
4732:
4731:
4710:
4709:
4700:
4699:
4533:
4532:
4422:
4421:
4329:
4328:
4266:
4265:
4221:
4220:
4194:
4193:
4171:, are given by
4170:
4168:
4167:
4162:
4160:
4156:
4155:
4154:
4107:
4106:
4075:
4073:
4072:
4067:
3994:
3992:
3991:
3986:
3951:
3950:
3920:
3919:
3895:
3894:
3876:
3875:
3829:
3827:
3826:
3821:
3819:
3818:
3800:
3799:
3777:
3773:
3771:
3770:
3765:
3763:
3762:
3744:
3743:
3727:
3723:
3719:
3712:
3710:
3709:
3704:
3653:
3645:
3606:
3604:
3603:
3598:
3593:
3589:
3567:
3566:
3533:
3532:
3520:
3518:
3507:
3496:
3494:
3489:
3478:
3465:
3464:
3463:
3460:
3411:
3399:
3397:
3396:
3391:
3389:
3375:
3374:
3353:
3352:
3330:
3319:
3310:
3309:
3308:
3295:
3267:
3236:
3235:
3214:
3135:greatest integer
3128:
3126:
3125:
3120:
3099:
3097:
3096:
3091:
3086:
3085:
3058:
3057:
3048:
3047:
3046:
3033:
3025:
3014:
2929:
2927:
2926:
2921:
2913:
2912:
2885:
2884:
2875:
2874:
2873:
2860:
2804:
2789:
2787:
2786:
2781:
2725:
2721:
2709:
2698:
2686:
2676:
2666:
2645:
2643:
2642:
2637:
2571:
2564:
2562:
2561:
2556:
2554:
2552:
2520:
2500:
2495:
2493:
2467:
2435:
2423:
2419:
2415:
2411:
2389:
2387:
2386:
2381:
2303:
2292:
2282:
2280:
2279:
2274:
2272:
2271:
2244:
2243:
2234:
2233:
2232:
2219:
2184:
2182:
2181:
2176:
2174:
2173:
2172:
2159:
2146:) must be added
2145:
2132:
2128:
2124:
2122:
2121:
2116:
2114:
2113:
2112:
2099:
2085:
2083:
2082:
2077:
2075:
2074:
2073:
2060:
2046:
2036:
2032:
2028:
2018:
2014:
2010:
1994:
1992:
1991:
1986:
1984:
1982:
1956:
1948:
1943:
1942:
1941:
1928:
1911:
1907:= 0, 1, 2, ...,
1898:
1896:
1895:
1890:
1888:
1887:
1860:
1859:
1850:
1849:
1848:
1835:
1766:
1762:
1758:
1754:
1735:
1734:
1725:
1723:
1721:
1720:
1715:
1713:
1700:
1695:
1675:
1671:
1663:
1658:with replacement
1655:
1636:
1621:
1607:
1588:, each asking a
1581:
1573:
1569:
1566:with parameters
1549:
1545:
1538:
1521:
1517:
1512:
1495:
1488:
1481:
1271:Elementary event
1203:
1181:
1180:
1174:
1172:
1171:
1166:
1153:
1151:
1150:
1145:
1143:
1141:
1130:
1116:
1115:
1091:
1089:
1088:
1083:
1081:
1080:
1026:
1024:
1023:
1018:
1016:
1015:
1006:
1005:
966:
964:
963:
958:
956:
955:
946:
945:
899:
897:
896:
891:
889:
885:
877:
838:
837:
828:
820:
803:
801:
800:
795:
793:
791:
780:
763:
746:
744:
743:
738:
736:
724:
723:
712:
695:
693:
692:
687:
637:
635:
634:
629:
590:
588:
587:
582:
541:
539:
538:
533:
512:
510:
509:
504:
475:
473:
472:
467:
439:
437:
436:
431:
390:
389:
365:
363:
362:
357:
355:
354:
339:
338:
329:
328:
327:
314:
291:
289:
288:
283:
233:
231:
230:
225:
200:
198:
197:
192:
161:
159:
158:
153:
103:
101:
100:
95:
62:
50:
40:
36:
21:
18874:
18873:
18869:
18868:
18867:
18865:
18864:
18863:
18834:
18833:
18832:
18827:
18799:
18775:Maximum entropy
18733:
18721:
18709:
18699:
18691:
18674:
18662:
18650:
18605:
18592:
18529:Matrix-valued:
18526:
18472:
18443:
18435:
18424:
18412:
18403:
18393:
18287:
18281:
18198:
18124:
18122:
18116:
18045:Maxwell–Jüttner
17894:Hypoexponential
17800:
17798:
17797:supported on a
17792:
17753:Noncentral beta
17713:Balding–Nichols
17695:
17694:supported on a
17686:
17676:
17579:
17573:
17569:Zipf–Mandelbrot
17499:
17490:
17484:
17474:
17406:
17396:
17367:
17365:Further reading
17362:
17361:
17351:
17349:
17345:
17340:
17332:
17328:
17321:
17307:
17303:
17272:
17268:
17255:
17251:
17242:
17241:
17237:
17230:
17216:
17212:
17206:9781-43983-5746
17199:
17195:
17179:
17178:
17172:
17170:
17165:
17164:
17160:
17141:
17134:
17127:
17113:
17109:
17093:
17092:
17086:
17084:
17079:
17078:
17074:
17056:
17052:
17036:
17035:
17029:
17027:
17022:
17021:
17017:
17010:
16996:
16992:
16985:
16963:
16956:
16946:
16944:
16936:Taboga, Marco.
16934:
16930:
16915:10.2307/2530610
16899:
16895:
16887:
16872:
16866:
16862:
16855:
16839:
16835:
16826:
16824:
16815:
16814:
16810:
16802:
16800:
16796:
16782:10.2307/2276774
16765:
16759:
16755:
16747:
16736:
16730:
16726:
16716:
16714:
16706:
16705:
16701:
16691:
16689:
16679:
16675:
16667:
16665:
16655:10.2307/2685469
16638:
16632:
16628:
16620:
16618:
16607:10.1.1.323.7752
16586:
16579:
16550:
16546:
16535:Marko Lalovic (
16534:
16530:
16487:
16483:
16475:
16464:
16458:
16454:
16447:
16425:
16421:
16382:
16375:
16326:
16322:
16299:
16295:
16272:
16265:
16256:
16252:
16237:
16233:
16215:
16211:
16179:
16175:
16141:
16137:
16103:
16099:
16091:
16087:
16072:10.2307/2986663
16056:
16052:
16033:
16029:
16010:
16006:
15999:
15985:
15981:
15977:
15964:Piling-up lemma
15919:
15912:
15909:
15861:Jacob Bernoulli
15857:
15844:
15840:
15836:
15835:for all values
15824:
15821:random variates
15809:
15803:
15798:
15789:
15785:
15781:
15739:
15726:
15722:
15698:
15694:
15693:
15691:
15662:
15659:
15658:
15559:
15556:
15555:
15545:
15530:
15519:
15515:
15511:
15507:
15500:
15469:approaches the
15414:
15412:
15410:
15407:
15406:
15344:
15285:
15243:
15240:
15232:
15229:
15228:
15191:hypothesis test
15118:
15117:
15085:
15080:
15076:
15061:
15040:
15038:
15034:
15023:
15020:
15019:
14994:
14991:
14990:
14974:
14971:
14970:
14914:
14878:
14875:
14874:
14843:
14840:
14839:
14808:
14805:
14804:
14785:
14782:
14781:
14743:
14740:
14739:
14720:
14717:
14716:
14712:
14711:
14684:
14661:
14655:
14635:
14632:
14631:
14627:
14625:
14622:
14621:
14589:
14569:
14563:
14543:
14540:
14525:
14523:
14520:
14519:
14482:
14476:
14461:
14455:
14431:
14428:
14413:
14411:
14408:
14407:
14382:
14379:
14378:
14344:
14339:
14335:
14320:
14299:
14297:
14293:
14282:
14279:
14278:
14258:
14254:
14237:
14234:
14233:
14216:
14212:
14207:
14204:
14203:
14175:
14172:
14171:
14129:
14099:
14073:
14059:
14056:
14055:
14004:
13986:
13954:
13940:
13937:
13936:
13879:
13865:
13862:
13861:
13823:
13818:
13814:
13799:
13778:
13776:
13772:
13761:
13758:
13757:
13694:
13665:
13662:
13661:
13605:
13599:
13579:
13576:
13575:
13571:
13559:
13525:
13508:
13507:
13505:
13503:
13500:
13499:
13486:
13398:
13397:
13395:
13392:
13391:
13346:
13340:
13284:
13279:
13278:
13237:
13234:
13233:
13216:
13215:
13203:
13199:
13178:
13174:
13158:
13145:
13144:
13143:
13134:
13133:
13121:
13117:
13084:
13080:
13064:
13051:
13050:
13049:
13042:
13020:
13018:
13015:
13014:
12984:
12980:
12951:
12948:
12947:
12910:
12906:
12888:
12884:
12862:
12847:
12841:
12840:
12839:
12827:
12816:
12811:
12807:
12801:
12797:
12781:
12768:
12767:
12766:
12743:
12740:
12739:
12708:
12705:
12704:
12687:
12686:
12669:
12665:
12641:
12617:
12613:
12612:
12605:
12593:
12582:
12576:
12575:
12574:
12568:
12557:
12552:
12548:
12542:
12538:
12522:
12509:
12508:
12507:
12498:
12497:
12480:
12476:
12452:
12448:
12424:
12420:
12413:
12401:
12390:
12384:
12383:
12382:
12376:
12365:
12360:
12356:
12350:
12346:
12340:
12336:
12329:
12316:
12315:
12314:
12307:
12285:
12283:
12280:
12279:
12260:
12257:
12256:
12233:
12229:
12223:
12219:
12210:
12206:
12204:
12201:
12200:
12174:
12170:
12146:
12142:
12124:
12120:
12114:
12110:
12103:
12091:
12080:
12074:
12073:
12072:
12065:
12052:
12051:
12050:
12044:
12033:
12006:
12003:
12002:
11977:
11965:
11954:
11948:
11947:
11945:
11937:
11924:
11923:
11921:
11910:
11897:
11896:
11894:
11886:
11873:
11872:
11870:
11868:
11865:
11864:
11847:
11846:
11834:
11830:
11806:
11802:
11784:
11780:
11774:
11770:
11763:
11750:
11749:
11748:
11741:
11728:
11727:
11726:
11720:
11709:
11696:
11695:
11641:
11630:
11619:
11597:
11595:
11592:
11591:
11547:
11544:
11543:
11506:
11503:
11502:
11488: ~ B(
11476: ~ B(
11470:
11463:
11452:
11439: ~ B(
11433:
11422:
11415:
11395: ~ B(
11371: ~ B(
11355: ~ B(
11349:
11337:
11326:
11319:
11318:) and variance
11317:
11310:
11275:
11271:
11257:
11253:
11239:
11233:
11225:
11206:
11173:
11172:
11152:
11149:
11148:
11147:distributed as
11120: ~ B(
11100: ~ B(
11088: ~ B(
11072: ~ B(
11052:
11051:
11033:
11029:
11011:
11007:
11000:
10985:
10979:
10978:
10977:
10968:
10967:
10944:
10940:
10916:
10912:
10905:
10893:
10884:
10883:
10882:
10881:
10877:
10860:
10856:
10838:
10834:
10827:
10814:
10813:
10812:
10811:
10807:
10801:
10790:
10779:
10754:
10752:
10749:
10748:
10736: ~ B(
10712: ~ B(
10700: ~ B(
10694:
10689:
10674:Clopper–Pearson
10670:
10638:
10634:
10632:
10625:
10613:
10609:
10605:
10599:
10595:
10593:
10569:
10567:
10566:
10545:
10543:
10542:
10541:
10539:
10537:
10521:
10515:
10511:
10509:
10494:
10492:
10491:
10490:
10488:
10486:
10483:
10482:
10453:
10449:
10436:
10426:
10422:
10414:
10411:
10410:
10384:
10380:
10367:
10363:
10359:
10351:
10348:
10347:
10331:
10328:
10327:
10306:
10296:
10292:
10284:
10281:
10280:
10259:
10255:
10251:
10243:
10240:
10239:
10235:)-th quantile'.
10226:
10212:
10206:
10172:
10168:
10163:
10143:
10141:
10139:
10135:
10128:
10124:
10115:
10111:
10109:
10106:
10105:
10100:
10094:
10060:
10056:
10054:
10051:
10050:
10028:
10025:
10024:
10023:. See here for
9995:
9991:
9989:
9986:
9985:
9963:
9960:
9959:
9936:
9932:
9925:
9918:
9914:
9904:
9895:
9891:
9890:
9888:
9874:
9873:
9871:
9868:
9867:
9863:is modified to
9836:
9832:
9825:
9810:
9809:
9789:
9788:
9787:
9784:
9767:
9766:
9764:
9761:
9760:
9755:
9749:
9697:
9695:
9694:
9673:
9671:
9670:
9669:
9666:
9648:
9646:
9645:
9643:
9640:
9639:
9636:
9630:
9608:
9605:
9604:
9578:
9570:
9567:
9566:
9550:
9547:
9546:
9526:
9523:
9522:
9485:
9477:
9474:
9473:
9457:
9454:
9453:
9429:
9425:
9423:
9408:
9406:
9405:
9403:
9400:
9399:
9391:
9373:
9367:
9338:
9329:
9318:
9317:
9316:
9314:
9311:
9310:
9269:
9264:
9255:
9244:
9243:
9242:
9240:
9237:
9236:
9216:
9205:
9204:
9203:
9201:
9198:
9197:
9166:
9165:
9163:
9160:
9159:
9121:
9110:
9103:
9101:
9071:
9060:
9059:
9058:
9056:
9053:
9052:
9025:
9006:
8989:
8986:
8985:
8984:, the prior is
8936:
8925:
8923:
8914:
8903:
8902:
8901:
8899:
8896:
8895:
8825:
8822:
8821:
8750:
8739:
8737:
8728:
8717:
8716:
8715:
8713:
8710:
8709:
8668:
8665:
8664:
8646:Bayes estimator
8617:, proven using
8575:
8561:
8560:
8558:
8555:
8554:
8540:
8534:
8529:
8498:
8483:
8470:
8469:
8465:
8464:
8454:
8450:
8434:
8416:
8396:
8393:
8392:
8330:
8329:
8325:
8315:
8311:
8290:
8261:
8258:
8257:
8211:
8210:
8206:
8196:
8192:
8168:
8147:
8135:
8106:
8103:
8102:
8058:
8055:
8054:
8016:
8005:
8003:
7969:
7931:
7928:
7927:
7852:
7851:
7847:
7837:
7833:
7798:
7795:
7794:
7719:
7704:
7697:
7693:
7692:
7682:
7678:
7643:
7640:
7639:
7552:
7549:
7548:
7486:
7483:
7482:
7471:
7444:
7436:
7433:
7432:
7431:is even, then
7407:
7399:
7396:
7395:
7378:
7377:
7362:
7361:
7351:
7336:
7335:
7320:
7319:
7309:
7307:
7304:
7303:
7274:
7266:
7263:
7262:
7234:
7226:
7223:
7222:
7190:
7182:
7179:
7178:
7135:
7118:
7116:
7113:
7112:
7045:
7031:
7014:
7012:
7009:
7008:
6955:
6952:
6951:
6925:
6922:
6921:
6902:
6899:
6898:
6887:
6806:
6803:
6802:
6786:
6757:
6754:
6753:
6722:
6719:
6718:
6681:
6678:
6677:
6640:
6637:
6636:
6619:
6618:
6549:
6548:
6479:
6478:
6408:
6406:
6403:
6402:
6350:
6330:
6328:
6309:
6289:
6287:
6285:
6282:
6281:
6250:
6247:
6246:
6221:
6218:
6217:
6201:
6198:
6197:
6175:
6172:
6171:
6149:
6146:
6145:
6114:
6111:
6110:
6079:
6076:
6075:
6053:
6050:
6049:
6018:
6015:
6014:
5989:
5986:
5985:
5963:
5960:
5959:
5930:
5926:
5920:
5916:
5909:
5896:
5895:
5894:
5877:
5874:
5873:
5847:
5846:
5811:
5809:
5803:
5802:
5752:
5750:
5719: and
5717:
5692:
5691:
5683:
5660:
5658:
5626:
5625:
5617:
5615:
5612:
5611:
5595: + 1)
5587: + 1)
5575: + 1)
5546:
5543:
5542:
5505:
5502:
5501:
5482:
5461:
5457:
5443:
5440:
5439:
5419:
5415:
5404:
5401:
5400:
5376:
5362:
5359:
5358:
5322:
5316:
5312:
5310:
5306:
5294:
5290:
5272:
5239:
5226:
5221:
5217:
5216:
5204:
5200:
5189:
5186:
5185:
5160:
5157:
5156:
5136:
5133:
5132:
5065:
5061:
5059:
5056:
5055:
5025:
5021:
5018:
5015:
5014:
4991:
4987:
4976:
4972:
4958:
4954:
4948:
4937:
4921:
4917:
4906:
4903:
4902:
4898:and in general
4882:
4881:
4872:
4868:
4862:
4858:
4827:
4818:
4814:
4802:
4801:
4785:
4766:
4764:
4761:
4760:
4740:
4739:
4727:
4723:
4705:
4701:
4695:
4691:
4534:
4528:
4524:
4521:
4520:
4423:
4417:
4413:
4410:
4409:
4330:
4324:
4320:
4317:
4316:
4267:
4261:
4257:
4254:
4253:
4222:
4216:
4212:
4209:
4208:
4195:
4189:
4185:
4181:
4179:
4176:
4175:
4150:
4146:
4121:
4117:
4102:
4098:
4096:
4093:
4092:
4089:central moments
4085:
4010:
4007:
4006:
3946:
3942:
3915:
3911:
3890:
3886:
3871:
3867:
3838:
3835:
3834:
3814:
3810:
3795:
3791:
3783:
3780:
3779:
3775:
3758:
3754:
3739:
3735:
3733:
3730:
3729:
3725:
3721:
3717:
3674:
3671:
3670:
3651:
3628:
3625:
3620:
3562:
3558:
3528:
3524:
3508:
3497:
3495:
3479:
3477:
3470:
3466:
3459:
3455:
3451:
3422:
3419:
3418:
3409:
3387:
3386:
3370:
3366:
3336:
3332:
3320:
3315:
3304:
3291:
3290:
3289:
3265:
3264:
3225:
3221:
3212:
3211:
3186:
3158:
3156:
3153:
3152:
3108:
3105:
3104:
3075:
3071:
3053:
3049:
3042:
3029:
3028:
3027:
3015:
3004:
2950:
2947:
2946:
2936:
2902:
2898:
2880:
2876:
2869:
2856:
2855:
2854:
2825:
2822:
2821:
2811:
2795:
2742:
2739:
2738:
2723:
2711:
2700:
2696:
2691: + 1)
2678:
2668:
2649:
2580:
2577:
2576:
2572:that satisfies
2569:
2521:
2501:
2499:
2468:
2436:
2434:
2432:
2429:
2428:
2421:
2417:
2413:
2394:
2312:
2309:
2308:
2294:
2287:
2261:
2257:
2239:
2235:
2228:
2215:
2214:
2213:
2190:
2187:
2186:
2168:
2155:
2154:
2153:
2151:
2148:
2147:
2138:
2130:
2126:
2108:
2095:
2094:
2093:
2091:
2088:
2087:
2069:
2056:
2055:
2054:
2052:
2049:
2048:
2038:
2037:successes (and
2034:
2030:
2020:
2016:
2012:
2003:
1957:
1949:
1947:
1937:
1924:
1923:
1922:
1920:
1917:
1916:
1903:
1877:
1873:
1855:
1851:
1844:
1831:
1830:
1829:
1779:
1776:
1775:
1764:
1760:
1756:
1737:
1732:
1727:
1709:
1707:
1704:
1703:
1698:
1697:
1693:
1691:random variable
1687:
1682:
1673:
1669:
1661:
1653:
1631:
1624:Bernoulli trial
1613:
1605:
1590:yes–no question
1579:
1571:
1567:
1547:
1540:
1533:
1532:with 8 layers (
1527:
1526:
1519:
1515:
1513:
1510:
1499:
1347:Random variable
1298:Bernoulli trial
1160:
1157:
1156:
1154:
1134:
1129:
1111:
1107:
1105:
1102:
1101:
1076:
1072:
1040:
1037:
1036:
1011:
1007:
998:
994:
980:
977:
976:
951:
947:
941:
937:
923:
920:
919:
900:
876:
872:
833:
829:
819:
817:
814:
813:
781:
764:
762:
760:
757:
756:
752:Excess kurtosis
713:
711:
709:
706:
705:
651:
648:
647:
596:
593:
592:
555:
552:
551:
518:
515:
514:
489:
486:
485:
458:
455:
454:
385:
381:
379:
376:
375:
344:
340:
334:
330:
323:
310:
309:
308:
306:
303:
302:
247:
244:
243:
207:
204:
203:
202:
168:
165:
164:
163:
117:
114:
113:
74:
71:
70:
57:
45:
35:
28:
23:
22:
15:
12:
11:
5:
18872:
18862:
18861:
18856:
18851:
18846:
18829:
18828:
18826:
18825:
18815:
18804:
18801:
18800:
18798:
18797:
18792:
18787:
18782:
18777:
18772:
18770:Location–scale
18767:
18762:
18757:
18752:
18747:
18741:
18739:
18735:
18734:
18732:
18731:
18726:
18719:
18714:
18706:
18704:
18693:
18692:
18690:
18689:
18684:
18679:
18672:
18667:
18660:
18655:
18648:
18643:
18638:
18633:
18631:Wrapped Cauchy
18628:
18626:Wrapped normal
18623:
18618:
18613:
18602:
18600:
18594:
18593:
18591:
18590:
18589:
18588:
18583:
18581:Normal-inverse
18578:
18573:
18563:
18562:
18561:
18551:
18543:
18538:
18533:
18524:
18523:
18522:
18512:
18504:
18499:
18494:
18489:
18488:
18487:
18477:
18470:
18469:
18468:
18463:
18453:
18448:
18440:
18438:
18430:
18429:
18426:
18425:
18423:
18422:
18416:
18414:
18405:
18399:
18398:
18395:
18394:
18392:
18391:
18386:
18381:
18373:
18365:
18357:
18348:
18339:
18330:
18321:
18312:
18307:
18302:
18297:
18291:
18289:
18283:
18282:
18280:
18279:
18274:
18272:Variance-gamma
18269:
18264:
18256:
18251:
18246:
18241:
18236:
18231:
18223:
18218:
18217:
18216:
18206:
18201:
18196:
18190:
18185:
18180:
18175:
18170:
18165:
18160:
18152:
18147:
18139:
18134:
18128:
18126:
18118:
18117:
18115:
18114:
18112:Wilks's lambda
18109:
18108:
18107:
18097:
18092:
18087:
18082:
18077:
18072:
18067:
18062:
18057:
18052:
18050:Mittag-Leffler
18047:
18042:
18037:
18032:
18027:
18022:
18017:
18012:
18007:
18002:
17997:
17992:
17991:
17990:
17980:
17971:
17966:
17961:
17960:
17959:
17949:
17947:gamma/Gompertz
17944:
17943:
17942:
17937:
17927:
17922:
17917:
17916:
17915:
17903:
17902:
17901:
17896:
17891:
17881:
17880:
17879:
17869:
17864:
17859:
17858:
17857:
17856:
17855:
17845:
17835:
17830:
17825:
17820:
17815:
17810:
17804:
17802:
17799:semi-infinite
17794:
17793:
17791:
17790:
17785:
17780:
17775:
17770:
17765:
17760:
17755:
17750:
17745:
17740:
17735:
17730:
17725:
17720:
17715:
17710:
17705:
17699:
17697:
17688:
17682:
17681:
17678:
17677:
17675:
17674:
17669:
17664:
17659:
17654:
17649:
17644:
17639:
17634:
17629:
17624:
17619:
17614:
17609:
17604:
17599:
17594:
17589:
17583:
17581:
17578:with infinite
17575:
17574:
17572:
17571:
17566:
17561:
17556:
17551:
17546:
17541:
17540:
17539:
17532:Hypergeometric
17529:
17524:
17519:
17514:
17509:
17503:
17501:
17492:
17486:
17485:
17473:
17472:
17465:
17458:
17450:
17444:
17443:
17433:
17428:
17418:
17413:
17405:
17404:External links
17402:
17401:
17400:
17394:
17381:
17366:
17363:
17360:
17359:
17342:
17341:
17339:
17338:
17326:
17319:
17301:
17282:(2): 216–222.
17266:
17249:
17235:
17229:978-0521642989
17228:
17210:
17193:
17158:
17132:
17125:
17107:
17072:
17050:
17015:
17008:
16990:
16983:
16954:
16928:
16909:(3): 469–474.
16893:
16890:on 2016-03-03.
16860:
16853:
16833:
16808:
16753:
16724:
16699:
16673:
16649:(2): 119–126,
16626:
16600:(2): 101–133,
16577:
16564:(2): 326–332.
16544:
16528:
16501:(3): 527–535.
16481:
16452:
16445:
16419:
16392:(1): 125–131.
16373:
16320:
16293:
16263:
16250:
16231:
16225:Stack Exchange
16209:
16173:
16155:(1): 101–103,
16135:
16117:(1): 197–204,
16097:
16085:
16050:
16027:
16004:
15997:
15978:
15976:
15973:
15972:
15971:
15961:
15956:
15946:
15941:
15936:
15931:
15925:
15924:
15908:
15905:
15856:
15853:
15802:
15799:
15797:
15794:
15778:
15777:
15766:
15760:
15757:
15754:
15751:
15748:
15745:
15742:
15735:
15732:
15729:
15725:
15721:
15718:
15715:
15712:
15707:
15704:
15701:
15697:
15690:
15687:
15684:
15681:
15678:
15675:
15672:
15669:
15666:
15641:
15640:
15629:
15626:
15623:
15620:
15617:
15614:
15611:
15608:
15605:
15602:
15599:
15596:
15593:
15590:
15587:
15584:
15581:
15578:
15575:
15572:
15569:
15566:
15563:
15499:
15496:
15495:
15494:
15466:
15465:
15464:
15463:
15449:
15446:
15443:
15440:
15437:
15434:
15431:
15426:
15423:
15420:
15417:
15401:
15400:
15384:
15378:expected value
15343:
15340:
15284:
15281:
15280:
15279:
15265:
15261:
15258:
15255:
15252:
15249:
15246:
15239:
15236:
15177:) is a sum of
15116:
15115:
15104:
15100:
15094:
15091:
15088:
15084:
15079:
15075:
15072:
15069:
15058:
15053:
15049:
15046:
15043:
15037:
15033:
15030:
15027:
15004:
15001:
14998:
14978:
14967:
14966:
14955:
14952:
14949:
14946:
14943:
14940:
14937:
14934:
14931:
14928:
14925:
14922:
14912:
14909:
14906:
14903:
14900:
14897:
14894:
14891:
14888:
14885:
14882:
14859:
14856:
14853:
14850:
14847:
14827:
14824:
14821:
14818:
14815:
14812:
14792:
14789:
14778:
14777:
14776:
14775:
14762:
14759:
14756:
14753:
14750:
14747:
14727:
14724:
14710:
14709:
14698:
14693:
14689:
14683:
14679:
14670:
14667:
14664:
14660:
14654:
14648:
14644:
14641:
14638:
14630:
14615:
14614:
14603:
14598:
14594:
14588:
14585:
14578:
14575:
14572:
14568:
14562:
14556:
14552:
14549:
14546:
14539:
14534:
14530:
14513:
14512:
14501:
14498:
14491:
14488:
14485:
14481:
14475:
14470:
14466:
14453:
14450:
14444:
14440:
14437:
14434:
14427:
14422:
14418:
14392:
14389:
14386:
14375:
14374:
14363:
14359:
14353:
14350:
14347:
14343:
14338:
14334:
14331:
14328:
14317:
14312:
14308:
14305:
14302:
14296:
14292:
14289:
14286:
14261:
14257:
14253:
14250:
14247:
14244:
14241:
14219:
14215:
14211:
14191:
14188:
14185:
14182:
14179:
14168:
14167:
14156:
14151:
14148:
14145:
14142:
14139:
14136:
14133:
14128:
14125:
14122:
14119:
14116:
14113:
14110:
14107:
14095:
14092:
14089:
14086:
14083:
14080:
14077:
14072:
14069:
14066:
14063:
14049:
14048:
14037:
14034:
14031:
14026:
14023:
14020:
14017:
14014:
14011:
14008:
14003:
14000:
13997:
13994:
13984:
13981:
13976:
13973:
13970:
13967:
13964:
13961:
13958:
13953:
13950:
13947:
13944:
13921:
13918:
13915:
13912:
13909:
13906:
13901:
13898:
13895:
13892:
13889:
13886:
13883:
13878:
13875:
13872:
13869:
13858:
13857:
13856:
13855:
13854:
13853:
13842:
13838:
13832:
13829:
13826:
13822:
13817:
13813:
13810:
13807:
13796:
13791:
13787:
13784:
13781:
13775:
13771:
13768:
13765:
13752:
13751:
13750:
13739:
13736:
13733:
13730:
13727:
13724:
13721:
13716:
13713:
13710:
13707:
13704:
13701:
13698:
13693:
13690:
13687:
13684:
13681:
13678:
13675:
13672:
13669:
13656:
13655:
13644:
13643:
13642:
13641:
13630:
13627:
13623:
13614:
13611:
13608:
13604:
13598:
13592:
13588:
13585:
13582:
13574:
13567:
13563:
13558:
13552:
13549:
13546:
13543:
13540:
13537:
13534:
13528:
13524:
13521:
13518:
13515:
13511:
13494:
13493:
13471:rules of thumb
13455:
13454:
13443:
13440:
13437:
13434:
13431:
13428:
13425:
13422:
13419:
13415:
13412:
13409:
13406:
13401:
13339:
13336:
13283:
13280:
13265:
13262:
13259:
13256:
13253:
13250:
13247:
13244:
13241:
13230:
13229:
13212:
13209:
13206:
13202:
13198:
13195:
13192:
13189:
13186:
13181:
13177:
13173:
13170:
13167:
13161:
13156:
13153:
13148:
13142:
13139:
13137:
13135:
13130:
13127:
13124:
13120:
13116:
13113:
13110:
13107:
13104:
13101:
13098:
13095:
13092:
13087:
13083:
13079:
13076:
13073:
13067:
13062:
13059:
13054:
13048:
13045:
13043:
13041:
13038:
13035:
13032:
13029:
13026:
13023:
13022:
12993:
12990:
12987:
12983:
12979:
12976:
12973:
12970:
12967:
12964:
12961:
12958:
12955:
12944:
12943:
12931:
12925:
12922:
12919:
12916:
12913:
12909:
12905:
12902:
12899:
12896:
12891:
12887:
12883:
12880:
12877:
12874:
12871:
12865:
12860:
12856:
12853:
12850:
12844:
12836:
12833:
12830:
12825:
12822:
12819:
12815:
12810:
12804:
12800:
12796:
12793:
12790:
12784:
12779:
12776:
12771:
12765:
12762:
12759:
12756:
12753:
12750:
12747:
12724:
12721:
12718:
12715:
12712:
12701:
12700:
12684:
12678:
12675:
12672:
12668:
12664:
12661:
12658:
12655:
12650:
12647:
12644:
12639:
12635:
12632:
12629:
12626:
12623:
12620:
12616:
12608:
12602:
12599:
12596:
12591:
12588:
12585:
12579:
12571:
12566:
12563:
12560:
12556:
12551:
12545:
12541:
12537:
12534:
12531:
12525:
12520:
12517:
12512:
12506:
12503:
12501:
12499:
12495:
12489:
12486:
12483:
12479:
12475:
12472:
12469:
12466:
12461:
12458:
12455:
12451:
12447:
12444:
12441:
12438:
12433:
12430:
12427:
12423:
12416:
12410:
12407:
12404:
12399:
12396:
12393:
12387:
12379:
12374:
12371:
12368:
12364:
12359:
12353:
12349:
12343:
12339:
12332:
12327:
12324:
12319:
12313:
12310:
12308:
12306:
12303:
12300:
12297:
12294:
12291:
12288:
12287:
12264:
12242:
12239:
12236:
12232:
12226:
12222:
12218:
12213:
12209:
12197:
12196:
12183:
12180:
12177:
12173:
12169:
12166:
12163:
12160:
12155:
12152:
12149:
12145:
12141:
12138:
12135:
12132:
12127:
12123:
12117:
12113:
12106:
12100:
12097:
12094:
12089:
12086:
12083:
12077:
12068:
12063:
12060:
12055:
12047:
12042:
12039:
12036:
12032:
12028:
12025:
12022:
12019:
12016:
12013:
12010:
11987:
11980:
11974:
11971:
11968:
11963:
11960:
11957:
11951:
11940:
11935:
11932:
11927:
11920:
11913:
11908:
11905:
11900:
11889:
11884:
11881:
11876:
11861:
11860:
11843:
11840:
11837:
11833:
11829:
11826:
11823:
11820:
11815:
11812:
11809:
11805:
11801:
11798:
11795:
11792:
11787:
11783:
11777:
11773:
11766:
11761:
11758:
11753:
11744:
11739:
11736:
11731:
11723:
11718:
11715:
11712:
11708:
11704:
11701:
11699:
11697:
11694:
11691:
11688:
11685:
11682:
11679:
11676:
11673:
11670:
11667:
11664:
11661:
11658:
11655:
11652:
11649:
11644:
11639:
11636:
11633:
11629:
11625:
11622:
11620:
11618:
11615:
11612:
11609:
11606:
11603:
11600:
11599:
11572:
11569:
11566:
11563:
11560:
11557:
11554:
11551:
11531:
11528:
11525:
11522:
11519:
11516:
11513:
11510:
11499:
11498:
11468:
11461:
11450:
11431:
11420:
11413:
11348:
11345:
11335:
11324:
11315:
11308:
11269:
11251:
11232:
11229:
11223:
11205:
11202:
11189:
11186:
11180:
11177:
11171:
11168:
11165:
11162:
11159:
11156:
11066:
11065:
11048:
11045:
11042:
11039:
11036:
11032:
11028:
11025:
11022:
11019:
11014:
11010:
11003:
10998:
10994:
10991:
10988:
10982:
10976:
10973:
10971:
10969:
10965:
10959:
10956:
10953:
10950:
10947:
10943:
10939:
10936:
10933:
10930:
10925:
10922:
10919:
10915:
10908:
10902:
10899:
10896:
10892:
10887:
10880:
10875:
10869:
10866:
10863:
10859:
10855:
10852:
10849:
10846:
10841:
10837:
10830:
10825:
10822:
10817:
10810:
10804:
10799:
10796:
10793:
10789:
10785:
10782:
10780:
10778:
10775:
10772:
10769:
10766:
10763:
10760:
10757:
10756:
10693:
10690:
10688:
10685:
10669:
10666:
10665:
10664:
10663:
10662:
10646:
10641:
10637:
10631:
10628:
10616:
10612:
10608:
10602:
10598:
10592:
10587:
10583:
10577:
10572:
10565:
10562:
10559:
10553:
10548:
10536:
10533:
10527:
10524:
10518:
10514:
10508:
10502:
10497:
10477:
10476:
10464:
10461:
10456:
10452:
10448:
10443:
10439:
10435:
10432:
10429:
10425:
10421:
10418:
10398:
10395:
10392:
10387:
10383:
10379:
10374:
10370:
10366:
10362:
10358:
10355:
10335:
10313:
10309:
10305:
10302:
10299:
10295:
10291:
10288:
10266:
10262:
10258:
10254:
10250:
10247:
10236:
10222:
10208:Main article:
10205:
10202:
10201:
10200:
10189:
10185:
10176:
10171:
10167:
10162:
10158:
10151:
10146:
10138:
10134:
10131:
10127:
10123:
10118:
10114:
10096:Main article:
10093:
10092:Arcsine method
10090:
10077:
10074:
10071:
10068:
10063:
10059:
10038:
10035:
10032:
10012:
10009:
10006:
10003:
9998:
9994:
9973:
9970:
9967:
9956:
9955:
9939:
9935:
9931:
9928:
9921:
9917:
9911:
9908:
9903:
9898:
9894:
9887:
9881:
9878:
9857:
9856:
9839:
9835:
9831:
9828:
9823:
9817:
9814:
9808:
9805:
9802:
9796:
9793:
9783:
9780:
9774:
9771:
9751:Main article:
9748:
9745:
9743:may be added.
9733:
9732:
9721:
9715:
9711:
9705:
9700:
9693:
9690:
9687:
9681:
9676:
9665:
9662:
9656:
9651:
9632:Main article:
9629:
9626:
9625:
9624:
9612:
9592:
9586:
9583:
9577:
9574:
9554:
9530:
9499:
9493:
9490:
9484:
9481:
9461:
9451:
9437:
9432:
9428:
9422:
9416:
9411:
9397:
9389:
9369:Main article:
9366:
9363:
9362:
9361:
9350:
9345:
9342:
9337:
9325:
9322:
9296:
9295:
9284:
9278:
9275:
9272:
9268:
9263:
9258:
9251:
9248:
9232:, leading to:
9219:
9212:
9209:
9185:
9182:
9179:
9173:
9170:
9148:
9147:
9136:
9130:
9127:
9124:
9117:
9114:
9109:
9106:
9100:
9095:
9092:
9089:
9086:
9083:
9080:
9077:
9074:
9067:
9064:
9037:
9032:
9029:
9024:
9021:
9018:
9013:
9010:
9005:
9002:
8999:
8996:
8993:
8982:Jeffreys prior
8967:posterior mode
8963:
8962:
8951:
8945:
8942:
8939:
8934:
8931:
8928:
8922:
8917:
8910:
8907:
8880:
8877:
8874:
8871:
8868:
8865:
8862:
8859:
8856:
8853:
8850:
8847:
8844:
8841:
8838:
8835:
8832:
8829:
8783:
8782:
8771:
8765:
8762:
8759:
8756:
8753:
8748:
8745:
8742:
8736:
8731:
8724:
8721:
8705:estimator is:
8703:posterior mean
8690:
8687:
8684:
8681:
8678:
8675:
8672:
8644:A closed form
8633:). It is also
8599:
8598:
8587:
8582:
8579:
8574:
8568:
8565:
8533:
8530:
8528:
8525:
8524:
8523:
8511:
8507:
8501:
8496:
8490:
8487:
8482:
8477:
8474:
8468:
8463:
8460:
8457:
8453:
8449:
8446:
8441:
8438:
8433:
8430:
8424:
8421:
8415:
8412:
8409:
8406:
8403:
8400:
8370:
8369:
8358:
8354:
8349:
8345:
8342:
8337:
8334:
8328:
8324:
8321:
8318:
8314:
8310:
8307:
8301:
8298:
8294:
8289:
8286:
8283:
8280:
8277:
8274:
8271:
8268:
8265:
8251:
8250:
8239:
8235:
8230:
8226:
8223:
8218:
8215:
8209:
8205:
8202:
8199:
8195:
8191:
8188:
8182:
8176:
8173:
8167:
8164:
8161:
8155:
8152:
8146:
8143:
8139:
8134:
8131:
8128:
8125:
8122:
8119:
8116:
8113:
8110:
8083:
8080:
8077:
8074:
8071:
8068:
8065:
8062:
8044:
8043:
8031:
8025:
8022:
8019:
8014:
8011:
8008:
8002:
7999:
7996:
7993:
7990:
7987:
7984:
7981:
7976:
7973:
7968:
7965:
7962:
7959:
7956:
7953:
7950:
7947:
7944:
7941:
7938:
7935:
7889:
7888:
7876:
7871:
7867:
7864:
7859:
7856:
7850:
7846:
7843:
7840:
7836:
7832:
7829:
7826:
7823:
7820:
7817:
7814:
7811:
7808:
7805:
7802:
7788:Chernoff bound
7745:
7744:
7732:
7728:
7722:
7717:
7711:
7708:
7703:
7700:
7696:
7691:
7688:
7685:
7681:
7677:
7674:
7671:
7668:
7665:
7662:
7659:
7656:
7653:
7650:
7647:
7610:
7607:
7604:
7601:
7598:
7595:
7592:
7589:
7586:
7583:
7580:
7577:
7574:
7571:
7568:
7565:
7562:
7559:
7556:
7532:
7529:
7526:
7523:
7520:
7517:
7514:
7511:
7508:
7505:
7502:
7499:
7496:
7493:
7490:
7470:
7467:
7466:
7465:
7451:
7448:
7443:
7440:
7414:
7411:
7406:
7403:
7381:
7376:
7373:
7370:
7365:
7358:
7355:
7350:
7347:
7344:
7339:
7334:
7331:
7328:
7323:
7316:
7313:
7281:
7278:
7273:
7270:
7259:
7241:
7238:
7233:
7230:
7215:
7197:
7194:
7189:
7186:
7166:
7163:
7160:
7157:
7154:
7151:
7148:
7145:
7142:
7138:
7134:
7131:
7128:
7125:
7121:
7097:
7085:
7082:
7079:
7076:
7073:
7070:
7067:
7064:
7061:
7058:
7054:
7051:
7048:
7044:
7041:
7038:
7034:
7030:
7027:
7024:
7021:
7017:
7001:
6989:
6986:
6983:
6980:
6977:
6974:
6971:
6968:
6965:
6962:
6959:
6944:
6932:
6929:
6909:
6906:
6886:
6883:
6870:
6867:
6864:
6861:
6858:
6855:
6852:
6849:
6846:
6843:
6840:
6837:
6834:
6831:
6828:
6825:
6822:
6819:
6816:
6813:
6810:
6789:
6785:
6782:
6779:
6776:
6773:
6770:
6767:
6764:
6761:
6741:
6738:
6735:
6732:
6729:
6726:
6706:
6703:
6700:
6697:
6694:
6691:
6688:
6685:
6665:
6662:
6659:
6656:
6653:
6650:
6647:
6644:
6633:
6632:
6617:
6614:
6611:
6608:
6605:
6602:
6599:
6596:
6593:
6590:
6587:
6584:
6581:
6578:
6575:
6572:
6569:
6566:
6563:
6560:
6557:
6554:
6551:
6550:
6547:
6544:
6541:
6538:
6535:
6532:
6529:
6526:
6523:
6520:
6517:
6514:
6511:
6508:
6505:
6502:
6499:
6496:
6493:
6490:
6487:
6484:
6481:
6480:
6477:
6474:
6471:
6468:
6465:
6462:
6459:
6456:
6453:
6450:
6447:
6444:
6441:
6438:
6435:
6432:
6429:
6426:
6423:
6420:
6417:
6414:
6411:
6410:
6396:
6395:
6380:
6377:
6374:
6371:
6368:
6365:
6362:
6359:
6356:
6353:
6348:
6345:
6342:
6339:
6336:
6333:
6327:
6321:
6318:
6315:
6312:
6307:
6304:
6301:
6298:
6295:
6292:
6266:
6263:
6260:
6257:
6254:
6231:
6228:
6225:
6205:
6185:
6182:
6179:
6159:
6156:
6153:
6133:
6130:
6127:
6124:
6121:
6118:
6098:
6095:
6092:
6089:
6086:
6083:
6063:
6060:
6057:
6037:
6034:
6031:
6028:
6025:
6022:
6002:
5999:
5996:
5993:
5973:
5970:
5967:
5956:
5955:
5944:
5939:
5936:
5933:
5929:
5923:
5919:
5912:
5907:
5904:
5899:
5893:
5890:
5887:
5884:
5881:
5864:
5863:
5850:
5845:
5842:
5839:
5836:
5833:
5830:
5827:
5824:
5821:
5818:
5810:
5808:
5805:
5804:
5801:
5798:
5795:
5792:
5789:
5786:
5783:
5780:
5777:
5774:
5771:
5768:
5765:
5762:
5759:
5751:
5749:
5746:
5743:
5739:
5736:
5733:
5730:
5727:
5713:
5709:
5706:
5703:
5700:
5697:
5694:
5693:
5690:
5682:
5679:
5676:
5673:
5670:
5667:
5659:
5657:
5654:
5650:
5647:
5644:
5641:
5638:
5635:
5632:
5631:
5629:
5624:
5569:floor function
5556:
5553:
5550:
5530:
5527:
5524:
5521:
5518:
5515:
5512:
5509:
5488:of a binomial
5481:
5478:
5464:
5460:
5456:
5453:
5450:
5447:
5427:
5422:
5418:
5414:
5411:
5408:
5388:
5383:
5380:
5375:
5372:
5369:
5366:
5355:
5354:
5353:
5352:
5341:
5337:
5331:
5328:
5325:
5319:
5315:
5309:
5305:
5302:
5297:
5293:
5289:
5286:
5283:
5280:
5275:
5270:
5264:
5261:
5258:
5255:
5252:
5249:
5246:
5242:
5238:
5235:
5232:
5229:
5225:
5220:
5215:
5212:
5207:
5203:
5199:
5196:
5193:
5164:
5140:
5120:
5117:
5114:
5111:
5108:
5105:
5102:
5099:
5096:
5093:
5090:
5087:
5084:
5081:
5078:
5072:
5069:
5064:
5037:
5032:
5029:
5024:
5011:
5010:
4999:
4994:
4990:
4983:
4980:
4975:
4970:
4965:
4962:
4957:
4951:
4946:
4943:
4940:
4936:
4932:
4929:
4924:
4920:
4916:
4913:
4910:
4896:
4895:
4880:
4875:
4871:
4865:
4861:
4857:
4854:
4851:
4848:
4845:
4842:
4839:
4836:
4833:
4830:
4828:
4826:
4821:
4817:
4813:
4810:
4807:
4804:
4803:
4800:
4797:
4794:
4791:
4788:
4786:
4784:
4781:
4778:
4775:
4772:
4769:
4768:
4754:
4753:
4738:
4735:
4730:
4726:
4722:
4719:
4716:
4713:
4708:
4704:
4698:
4694:
4690:
4687:
4684:
4681:
4678:
4675:
4672:
4669:
4666:
4663:
4660:
4657:
4654:
4651:
4648:
4645:
4642:
4639:
4636:
4633:
4630:
4627:
4624:
4621:
4618:
4615:
4612:
4609:
4606:
4603:
4600:
4597:
4594:
4591:
4588:
4585:
4582:
4579:
4576:
4573:
4570:
4567:
4564:
4561:
4558:
4555:
4552:
4549:
4546:
4543:
4540:
4537:
4535:
4531:
4527:
4523:
4522:
4519:
4516:
4513:
4510:
4507:
4504:
4501:
4498:
4495:
4492:
4489:
4486:
4483:
4480:
4477:
4474:
4471:
4468:
4465:
4462:
4459:
4456:
4453:
4450:
4447:
4444:
4441:
4438:
4435:
4432:
4429:
4426:
4424:
4420:
4416:
4412:
4411:
4408:
4405:
4402:
4399:
4396:
4393:
4390:
4387:
4384:
4381:
4378:
4375:
4372:
4369:
4366:
4363:
4360:
4357:
4354:
4351:
4348:
4345:
4342:
4339:
4336:
4333:
4331:
4327:
4323:
4319:
4318:
4315:
4312:
4309:
4306:
4303:
4300:
4297:
4294:
4291:
4288:
4285:
4282:
4279:
4276:
4273:
4270:
4268:
4264:
4260:
4256:
4255:
4252:
4249:
4246:
4243:
4240:
4237:
4234:
4231:
4228:
4225:
4223:
4219:
4215:
4211:
4210:
4207:
4204:
4201:
4198:
4196:
4192:
4188:
4184:
4183:
4159:
4153:
4149:
4145:
4142:
4139:
4136:
4133:
4130:
4127:
4124:
4120:
4116:
4113:
4110:
4105:
4101:
4084:
4083:Higher moments
4081:
4077:
4076:
4065:
4062:
4059:
4056:
4053:
4050:
4047:
4044:
4041:
4038:
4035:
4032:
4029:
4026:
4023:
4020:
4017:
4014:
3996:
3995:
3984:
3981:
3978:
3975:
3972:
3969:
3966:
3963:
3960:
3957:
3954:
3949:
3945:
3941:
3938:
3935:
3932:
3929:
3926:
3923:
3918:
3914:
3910:
3907:
3904:
3901:
3898:
3893:
3889:
3885:
3882:
3879:
3874:
3870:
3866:
3863:
3860:
3857:
3854:
3851:
3848:
3845:
3842:
3817:
3813:
3809:
3806:
3803:
3798:
3794:
3790:
3787:
3761:
3757:
3753:
3750:
3747:
3742:
3738:
3720:is the sum of
3714:
3713:
3702:
3699:
3696:
3693:
3690:
3687:
3684:
3681:
3678:
3660:expected value
3624:
3621:
3619:
3616:
3608:
3607:
3596:
3592:
3588:
3585:
3582:
3579:
3576:
3573:
3570:
3565:
3561:
3557:
3554:
3551:
3548:
3545:
3542:
3539:
3536:
3531:
3527:
3523:
3517:
3514:
3511:
3506:
3503:
3500:
3492:
3488:
3485:
3482:
3476:
3473:
3469:
3458:
3454:
3450:
3447:
3444:
3441:
3438:
3435:
3432:
3429:
3426:
3401:
3400:
3385:
3382:
3379:
3373:
3369:
3365:
3362:
3359:
3356:
3351:
3348:
3345:
3342:
3339:
3335:
3329:
3326:
3323:
3318:
3314:
3307:
3302:
3299:
3294:
3288:
3285:
3282:
3279:
3276:
3273:
3270:
3268:
3266:
3263:
3260:
3257:
3254:
3251:
3248:
3245:
3242:
3239:
3234:
3231:
3228:
3224:
3220:
3217:
3215:
3213:
3210:
3207:
3204:
3201:
3198:
3195:
3192:
3189:
3187:
3185:
3182:
3179:
3176:
3173:
3170:
3167:
3164:
3161:
3160:
3148:, as follows:
3118:
3115:
3112:
3101:
3100:
3089:
3084:
3081:
3078:
3074:
3070:
3067:
3064:
3061:
3056:
3052:
3045:
3040:
3037:
3032:
3024:
3021:
3018:
3013:
3010:
3007:
3003:
2999:
2996:
2993:
2990:
2987:
2984:
2981:
2978:
2975:
2972:
2969:
2966:
2963:
2960:
2957:
2954:
2935:
2932:
2931:
2930:
2919:
2916:
2911:
2908:
2905:
2901:
2897:
2894:
2891:
2888:
2883:
2879:
2872:
2867:
2864:
2859:
2853:
2850:
2847:
2844:
2841:
2838:
2835:
2832:
2829:
2810:
2807:
2792:floor function
2779:
2776:
2773:
2770:
2767:
2764:
2761:
2758:
2755:
2752:
2749:
2746:
2737:Equivalently,
2647:
2646:
2635:
2632:
2629:
2626:
2623:
2620:
2617:
2614:
2611:
2608:
2605:
2602:
2599:
2596:
2593:
2590:
2587:
2584:
2566:
2565:
2551:
2548:
2545:
2542:
2539:
2536:
2533:
2530:
2527:
2524:
2519:
2516:
2513:
2510:
2507:
2504:
2498:
2492:
2489:
2486:
2483:
2480:
2477:
2474:
2471:
2466:
2463:
2460:
2457:
2454:
2451:
2448:
2445:
2442:
2439:
2391:
2390:
2379:
2376:
2373:
2370:
2367:
2364:
2361:
2358:
2355:
2352:
2349:
2346:
2343:
2340:
2337:
2334:
2331:
2328:
2325:
2322:
2319:
2316:
2270:
2267:
2264:
2260:
2256:
2253:
2250:
2247:
2242:
2238:
2231:
2226:
2223:
2218:
2212:
2209:
2206:
2203:
2200:
2197:
2194:
2171:
2166:
2163:
2158:
2111:
2106:
2103:
2098:
2072:
2067:
2064:
2059:
1996:
1995:
1981:
1978:
1975:
1972:
1969:
1966:
1963:
1960:
1955:
1952:
1946:
1940:
1935:
1932:
1927:
1900:
1899:
1886:
1883:
1880:
1876:
1872:
1869:
1866:
1863:
1858:
1854:
1847:
1842:
1839:
1834:
1828:
1825:
1822:
1819:
1816:
1813:
1810:
1807:
1804:
1801:
1798:
1795:
1792:
1789:
1786:
1783:
1733:[0, 1]
1712:
1686:
1683:
1681:
1678:
1501:
1500:
1498:
1497:
1490:
1483:
1475:
1472:
1471:
1470:
1469:
1464:
1456:
1455:
1454:
1453:
1448:
1446:Bayes' theorem
1443:
1438:
1433:
1428:
1420:
1419:
1418:
1417:
1412:
1407:
1402:
1394:
1393:
1392:
1391:
1390:
1389:
1384:
1379:
1377:Observed value
1374:
1369:
1364:
1362:Expected value
1359:
1354:
1344:
1339:
1338:
1337:
1332:
1327:
1322:
1317:
1312:
1302:
1301:
1300:
1290:
1289:
1288:
1283:
1278:
1273:
1268:
1258:
1253:
1245:
1244:
1243:
1242:
1237:
1232:
1231:
1230:
1220:
1219:
1218:
1205:
1204:
1196:
1195:
1189:
1188:
1177:
1176:
1164:
1140:
1137:
1133:
1128:
1125:
1122:
1119:
1114:
1110:
1099:
1093:
1092:
1079:
1075:
1071:
1068:
1065:
1062:
1059:
1056:
1053:
1050:
1047:
1044:
1034:
1028:
1027:
1014:
1010:
1004:
1001:
997:
993:
990:
987:
984:
974:
968:
967:
954:
950:
944:
940:
936:
933:
930:
927:
917:
911:
910:
888:
883:
880:
875:
871:
868:
865:
862:
859:
856:
853:
850:
847:
844:
841:
836:
832:
826:
823:
811:
805:
804:
790:
787:
784:
779:
776:
773:
770:
767:
754:
748:
747:
733:
730:
727:
722:
719:
716:
703:
697:
696:
685:
682:
679:
676:
673:
670:
667:
664:
661:
658:
655:
645:
639:
638:
627:
624:
621:
618:
615:
612:
609:
606:
603:
600:
580:
577:
574:
571:
568:
565:
562:
559:
549:
543:
542:
531:
528:
525:
522:
502:
499:
496:
493:
483:
477:
476:
465:
462:
452:
446:
445:
429:
426:
423:
420:
417:
414:
411:
408:
405:
402:
399:
396:
393:
388:
384:
373:
367:
366:
353:
350:
347:
343:
337:
333:
326:
321:
318:
313:
300:
294:
293:
281:
278:
275:
272:
269:
266:
263:
260:
257:
254:
251:
241:
235:
234:
223:
220:
217:
214:
211:
190:
187:
184:
181:
178:
175:
172:
151:
148:
145:
142:
139:
136:
133:
130:
127:
124:
121:
111:
105:
104:
93:
90:
87:
84:
81:
78:
68:
64:
63:
55:
52:
51:
43:
26:
18:Binomial model
9:
6:
4:
3:
2:
18871:
18860:
18857:
18855:
18852:
18850:
18847:
18845:
18842:
18841:
18839:
18824:
18816:
18814:
18806:
18805:
18802:
18796:
18793:
18791:
18788:
18786:
18783:
18781:
18778:
18776:
18773:
18771:
18768:
18766:
18763:
18761:
18758:
18756:
18753:
18751:
18748:
18746:
18743:
18742:
18740:
18736:
18730:
18727:
18724:
18720:
18718:
18715:
18712:
18708:
18707:
18705:
18703:
18698:
18694:
18688:
18685:
18683:
18680:
18677:
18673:
18671:
18668:
18665:
18661:
18659:
18656:
18653:
18649:
18647:
18644:
18642:
18639:
18637:
18634:
18632:
18629:
18627:
18624:
18622:
18619:
18617:
18614:
18611:
18610:
18604:
18603:
18601:
18599:
18595:
18587:
18584:
18582:
18579:
18577:
18574:
18572:
18569:
18568:
18567:
18564:
18560:
18557:
18556:
18555:
18552:
18550:
18549:
18544:
18542:
18541:Matrix normal
18539:
18537:
18534:
18531:
18530:
18525:
18521:
18518:
18517:
18516:
18513:
18511:
18510:
18507:Multivariate
18505:
18503:
18500:
18498:
18495:
18493:
18490:
18486:
18483:
18482:
18481:
18478:
18475:
18471:
18467:
18464:
18462:
18459:
18458:
18457:
18454:
18452:
18449:
18446:
18442:
18441:
18439:
18437:
18434:Multivariate
18431:
18421:
18418:
18417:
18415:
18409:
18406:
18400:
18390:
18387:
18385:
18382:
18380:
18378:
18374:
18372:
18370:
18366:
18364:
18362:
18358:
18356:
18354:
18349:
18347:
18345:
18340:
18338:
18336:
18331:
18329:
18327:
18322:
18320:
18318:
18313:
18311:
18308:
18306:
18303:
18301:
18298:
18296:
18293:
18292:
18290:
18286:with support
18284:
18278:
18275:
18273:
18270:
18268:
18265:
18263:
18262:
18257:
18255:
18252:
18250:
18247:
18245:
18242:
18240:
18237:
18235:
18232:
18230:
18229:
18224:
18222:
18219:
18215:
18212:
18211:
18210:
18207:
18205:
18202:
18200:
18199:
18191:
18189:
18186:
18184:
18181:
18179:
18176:
18174:
18171:
18169:
18166:
18164:
18161:
18159:
18158:
18153:
18151:
18148:
18146:
18145:
18140:
18138:
18135:
18133:
18130:
18129:
18127:
18123:on the whole
18119:
18113:
18110:
18106:
18103:
18102:
18101:
18098:
18096:
18095:type-2 Gumbel
18093:
18091:
18088:
18086:
18083:
18081:
18078:
18076:
18073:
18071:
18068:
18066:
18063:
18061:
18058:
18056:
18053:
18051:
18048:
18046:
18043:
18041:
18038:
18036:
18033:
18031:
18028:
18026:
18023:
18021:
18018:
18016:
18013:
18011:
18008:
18006:
18003:
18001:
17998:
17996:
17993:
17989:
17986:
17985:
17984:
17981:
17979:
17977:
17972:
17970:
17967:
17965:
17964:Half-logistic
17962:
17958:
17955:
17954:
17953:
17950:
17948:
17945:
17941:
17938:
17936:
17933:
17932:
17931:
17928:
17926:
17923:
17921:
17920:Folded normal
17918:
17914:
17911:
17910:
17909:
17908:
17904:
17900:
17897:
17895:
17892:
17890:
17887:
17886:
17885:
17882:
17878:
17875:
17874:
17873:
17870:
17868:
17865:
17863:
17860:
17854:
17851:
17850:
17849:
17846:
17844:
17841:
17840:
17839:
17836:
17834:
17831:
17829:
17826:
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17809:
17806:
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17795:
17789:
17786:
17784:
17781:
17779:
17776:
17774:
17771:
17769:
17766:
17764:
17763:Raised cosine
17761:
17759:
17756:
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17751:
17749:
17746:
17744:
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17736:
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17663:
17660:
17658:
17655:
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17650:
17648:
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17640:
17638:
17637:Mixed Poisson
17635:
17633:
17630:
17628:
17625:
17623:
17620:
17618:
17615:
17613:
17610:
17608:
17605:
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17542:
17538:
17535:
17534:
17533:
17530:
17528:
17525:
17523:
17520:
17518:
17517:Beta-binomial
17515:
17513:
17510:
17508:
17505:
17504:
17502:
17496:
17493:
17487:
17482:
17478:
17471:
17466:
17464:
17459:
17457:
17452:
17451:
17448:
17442:
17439:available at
17438:
17434:
17432:
17429:
17427:
17423:
17419:
17417:
17414:
17412:
17408:
17407:
17397:
17395:0-205-10328-6
17391:
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17126:9789814288484
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17009:9789814288484
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16984:9780471093152
16980:
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16864:
16856:
16850:
16846:
16845:
16837:
16822:
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16812:
16799:on 2015-01-13
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16485:
16474:
16470:
16469:lecture notes
16463:
16456:
16448:
16446:9780486665214
16442:
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16270:
16268:
16260:
16254:
16246:
16243:(in German).
16242:
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16221:
16213:
16206:
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15893:Blaise Pascal
15890:
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11409:
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11383:, given
11382:
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11367: |
11366:
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11344:
11341:
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9323:
9320:
9309:
9308:
9307:
9305:
9304:rule of three
9301:
9282:
9276:
9273:
9270:
9266:
9261:
9256:
9249:
9246:
9235:
9234:
9233:
9217:
9210:
9207:
9183:
9180:
9177:
9171:
9168:
9157:
9153:
9134:
9128:
9125:
9122:
9115:
9112:
9107:
9104:
9098:
9093:
9090:
9087:
9084:
9081:
9078:
9075:
9072:
9065:
9062:
9051:
9050:
9049:
9030:
9027:
9022:
9019:
9016:
9011:
9008:
9003:
9000:
8994:
8991:
8983:
8978:
8976:
8972:
8968:
8949:
8943:
8940:
8937:
8932:
8929:
8926:
8920:
8915:
8908:
8905:
8894:
8893:
8892:
8875:
8872:
8869:
8863:
8860:
8854:
8851:
8848:
8845:
8842:
8839:
8836:
8830:
8827:
8819:
8815:
8810:
8808:
8804:
8800:
8796:
8792:
8788:
8769:
8763:
8760:
8757:
8754:
8751:
8746:
8743:
8740:
8734:
8729:
8722:
8719:
8708:
8707:
8706:
8704:
8685:
8682:
8679:
8673:
8670:
8662:
8659:
8655:
8651:
8647:
8642:
8640:
8636:
8632:
8628:
8624:
8620:
8616:
8612:
8608:
8605:and also the
8604:
8585:
8580:
8577:
8572:
8566:
8563:
8553:
8552:
8551:
8549:
8545:
8539:
8509:
8505:
8499:
8494:
8488:
8485:
8480:
8475:
8472:
8466:
8461:
8458:
8455:
8451:
8447:
8444:
8439:
8436:
8431:
8422:
8419:
8413:
8410:
8407:
8404:
8398:
8391:
8390:
8389:
8387:
8383:
8379:
8375:
8356:
8352:
8347:
8343:
8340:
8335:
8332:
8326:
8322:
8319:
8316:
8312:
8308:
8305:
8299:
8296:
8292:
8287:
8281:
8278:
8275:
8272:
8269:
8263:
8256:
8255:
8254:
8237:
8233:
8228:
8224:
8221:
8216:
8213:
8207:
8203:
8200:
8197:
8193:
8189:
8186:
8174:
8171:
8165:
8162:
8153:
8150:
8144:
8141:
8137:
8132:
8126:
8123:
8120:
8117:
8114:
8108:
8101:
8100:
8099:
8097:
8078:
8075:
8072:
8069:
8066:
8060:
8052:
8047:
8029:
8023:
8020:
8017:
8012:
8009:
8006:
8000:
7997:
7991:
7988:
7985:
7979:
7974:
7971:
7966:
7963:
7957:
7951:
7945:
7942:
7939:
7933:
7926:
7925:
7924:
7922:
7918:
7914:
7910:
7906:
7902:
7898:
7894:
7874:
7869:
7865:
7862:
7857:
7854:
7848:
7844:
7841:
7838:
7834:
7830:
7827:
7824:
7818:
7815:
7812:
7809:
7806:
7800:
7793:
7792:
7791:
7789:
7784:
7782:
7778:
7774:
7770:
7766:
7762:
7758:
7754:
7750:
7730:
7726:
7720:
7715:
7709:
7706:
7701:
7698:
7694:
7689:
7686:
7683:
7679:
7675:
7672:
7669:
7663:
7660:
7657:
7654:
7651:
7645:
7638:
7637:
7636:
7634:
7630:
7628:
7624:
7605:
7602:
7599:
7596:
7593:
7590:
7587:
7584:
7581:
7575:
7572:
7566:
7563:
7560:
7546:
7527:
7524:
7521:
7512:
7506:
7503:
7500:
7497:
7494:
7488:
7480:
7476:
7449:
7446:
7441:
7438:
7430:
7412:
7409:
7404:
7401:
7374:
7371:
7368:
7356:
7353:
7348:
7345:
7342:
7332:
7329:
7326:
7314:
7311:
7301:
7297:
7279:
7276:
7271:
7268:
7260:
7257:
7239:
7236:
7231:
7228:
7220:
7216:
7213:
7195:
7192:
7187:
7184:
7161:
7158:
7155:
7152:
7149:
7140:
7132:
7129:
7126:
7123:
7110:
7106:
7103: =
7102:
7098:
7077:
7074:
7071:
7068:
7065:
7056:
7052:
7049:
7046:
7036:
7028:
7025:
7022:
7019:
7006:
7002:
6984:
6981:
6975:
6972:
6969:
6963:
6960:
6949:
6945:
6930:
6927:
6907:
6904:
6896:
6895:
6894:
6892:
6882:
6865:
6859:
6856:
6853:
6844:
6841:
6838:
6832:
6829:
6826:
6820:
6817:
6814:
6783:
6780:
6777:
6774:
6768:
6765:
6762:
6739:
6733:
6730:
6727:
6704:
6701:
6698:
6692:
6689:
6686:
6663:
6660:
6657:
6651:
6648:
6645:
6612:
6606:
6603:
6597:
6594:
6591:
6585:
6579:
6576:
6573:
6567:
6564:
6561:
6555:
6552:
6542:
6536:
6533:
6527:
6524:
6521:
6515:
6509:
6506:
6503:
6497:
6494:
6491:
6485:
6482:
6472:
6466:
6463:
6457:
6454:
6451:
6445:
6439:
6436:
6433:
6427:
6424:
6421:
6415:
6412:
6401:
6400:
6399:
6375:
6372:
6369:
6360:
6357:
6354:
6346:
6340:
6337:
6334:
6325:
6316:
6310:
6302:
6299:
6296:
6290:
6280:
6279:
6278:
6264:
6261:
6258:
6255:
6252:
6243:
6229:
6226:
6223:
6203:
6183:
6180:
6177:
6157:
6154:
6151:
6131:
6128:
6122:
6116:
6096:
6093:
6087:
6081:
6061:
6058:
6055:
6035:
6032:
6026:
6020:
5997:
5991:
5971:
5968:
5965:
5942:
5937:
5934:
5931:
5927:
5921:
5917:
5905:
5902:
5891:
5885:
5879:
5872:
5871:
5870:
5868:
5843:
5840:
5837:
5834:
5831:
5825:
5822:
5819:
5806:
5799:
5793:
5790:
5787:
5784:
5781:
5775:
5772:
5766:
5763:
5760:
5747:
5744:
5741:
5734:
5731:
5728:
5711:
5704:
5701:
5698:
5688:
5680:
5674:
5671:
5668:
5652:
5645:
5642:
5639:
5627:
5622:
5610:
5609:
5608:
5606:
5602:
5598:
5594:
5590:
5586:
5582:
5578:
5574:
5570:
5551:
5525:
5519:
5516:
5513:
5499:
5495:
5491:
5487:
5477:
5462:
5454:
5448:
5420:
5416:
5409:
5381:
5378:
5370:
5367:
5364:
5339:
5335:
5329:
5326:
5323:
5317:
5313:
5307:
5303:
5300:
5295:
5287:
5284:
5278:
5273:
5268:
5259:
5256:
5250:
5247:
5240:
5236:
5230:
5227:
5223:
5218:
5213:
5205:
5201:
5194:
5184:
5183:
5182:
5181:
5180:
5178:
5162:
5154:
5153:falling power
5138:
5115:
5112:
5109:
5106:
5103:
5097:
5091:
5088:
5085:
5079:
5076:
5070:
5067:
5062:
5053:
5035:
5030:
5027:
5022:
4997:
4992:
4988:
4981:
4978:
4973:
4968:
4963:
4960:
4955:
4949:
4944:
4941:
4938:
4934:
4930:
4922:
4918:
4911:
4901:
4900:
4899:
4878:
4873:
4869:
4863:
4859:
4855:
4849:
4846:
4843:
4837:
4834:
4831:
4829:
4819:
4815:
4808:
4798:
4795:
4792:
4789:
4787:
4779:
4773:
4759:
4758:
4757:
4736:
4728:
4720:
4717:
4714:
4706:
4702:
4696:
4692:
4688:
4685:
4676:
4673:
4670:
4664:
4661:
4658:
4655:
4646:
4643:
4640:
4634:
4631:
4628:
4625:
4616:
4613:
4610:
4604:
4601:
4598:
4595:
4586:
4583:
4580:
4574:
4571:
4568:
4565:
4556:
4553:
4550:
4544:
4541:
4538:
4536:
4529:
4525:
4517:
4508:
4505:
4502:
4496:
4490:
4487:
4484:
4481:
4475:
4472:
4463:
4460:
4457:
4454:
4445:
4442:
4439:
4433:
4430:
4427:
4425:
4418:
4414:
4406:
4397:
4394:
4391:
4385:
4379:
4376:
4373:
4370:
4364:
4361:
4352:
4349:
4346:
4340:
4337:
4334:
4332:
4325:
4321:
4313:
4307:
4304:
4301:
4298:
4289:
4286:
4283:
4277:
4274:
4271:
4269:
4262:
4258:
4250:
4244:
4241:
4238:
4232:
4229:
4226:
4224:
4217:
4213:
4205:
4202:
4199:
4197:
4190:
4186:
4174:
4173:
4172:
4157:
4151:
4140:
4134:
4128:
4125:
4118:
4114:
4108:
4103:
4099:
4091:, defined as
4090:
4080:
4063:
4057:
4054:
4051:
4045:
4042:
4039:
4036:
4033:
4030:
4027:
4021:
4015:
4012:
4005:
4004:
4003:
4001:
3982:
3979:
3976:
3973:
3970:
3967:
3964:
3961:
3958:
3955:
3947:
3943:
3936:
3930:
3927:
3924:
3916:
3912:
3905:
3899:
3891:
3887:
3883:
3880:
3877:
3872:
3868:
3861:
3855:
3849:
3843:
3833:
3832:
3831:
3815:
3811:
3807:
3804:
3801:
3796:
3792:
3788:
3785:
3759:
3755:
3751:
3748:
3745:
3740:
3736:
3700:
3697:
3694:
3691:
3685:
3679:
3669:
3668:
3667:
3665:
3661:
3657:
3649:
3643:
3639:
3635:
3631:
3615:
3613:
3594:
3590:
3583:
3580:
3577:
3571:
3568:
3563:
3559:
3555:
3549:
3546:
3543:
3537:
3534:
3529:
3525:
3521:
3515:
3512:
3509:
3504:
3501:
3498:
3490:
3486:
3483:
3480:
3474:
3471:
3467:
3461:-distribution
3456:
3452:
3448:
3442:
3439:
3436:
3433:
3430:
3424:
3417:
3416:
3415:
3413:
3412:-distribution
3406:
3383:
3380:
3377:
3371:
3363:
3360:
3357:
3349:
3346:
3343:
3340:
3337:
3333:
3327:
3324:
3321:
3316:
3312:
3300:
3297:
3283:
3280:
3277:
3271:
3269:
3258:
3255:
3252:
3249:
3246:
3243:
3240:
3232:
3229:
3226:
3222:
3218:
3216:
3205:
3202:
3199:
3190:
3188:
3180:
3177:
3174:
3171:
3168:
3162:
3151:
3150:
3149:
3147:
3142:
3140:
3136:
3132:
3113:
3087:
3082:
3079:
3076:
3068:
3065:
3062:
3054:
3050:
3038:
3035:
3019:
3011:
3008:
3005:
3001:
2997:
2991:
2988:
2985:
2976:
2970:
2967:
2964:
2961:
2958:
2952:
2945:
2944:
2943:
2941:
2917:
2914:
2909:
2906:
2903:
2895:
2892:
2889:
2881:
2877:
2865:
2862:
2851:
2845:
2842:
2839:
2836:
2833:
2827:
2820:
2819:
2818:
2816:
2806:
2802:
2798:
2793:
2790:. Taking the
2777:
2774:
2771:
2768:
2765:
2762:
2759:
2756:
2753:
2750:
2747:
2744:
2735:
2733:
2729:
2728:most probable
2719:
2715:
2708:
2704:
2694:
2690:
2685:
2681:
2675:
2671:
2664:
2660:
2656:
2652:
2633:
2630:
2624:
2621:
2618:
2612:
2609:
2606:
2603:
2600:
2597:
2591:
2588:
2585:
2575:
2574:
2573:
2546:
2543:
2540:
2531:
2528:
2525:
2517:
2511:
2508:
2505:
2496:
2487:
2484:
2481:
2478:
2475:
2469:
2461:
2458:
2455:
2452:
2449:
2446:
2443:
2437:
2427:
2426:
2425:
2416:, there is a
2409:
2405:
2401:
2397:
2377:
2371:
2368:
2365:
2362:
2359:
2356:
2353:
2350:
2347:
2341:
2338:
2332:
2329:
2326:
2323:
2320:
2314:
2307:
2306:
2305:
2301:
2297:
2290:
2284:
2268:
2265:
2262:
2254:
2251:
2248:
2240:
2236:
2224:
2221:
2210:
2204:
2201:
2198:
2185:times, hence
2164:
2161:
2144:
2141:
2136:
2104:
2101:
2065:
2062:
2045:
2041:
2027:
2023:
2009:
2006:
2001:
1979:
1973:
1970:
1967:
1961:
1958:
1953:
1950:
1944:
1933:
1930:
1915:
1914:
1913:
1910:
1906:
1884:
1881:
1878:
1870:
1867:
1864:
1856:
1852:
1840:
1837:
1826:
1820:
1817:
1814:
1805:
1799:
1796:
1793:
1790:
1787:
1781:
1774:
1773:
1772:
1770:
1759:successes in
1752:
1748:
1744:
1740:
1730:
1724:
1692:
1677:
1667:
1659:
1650:
1648:
1644:
1643:binomial test
1640:
1634:
1629:
1625:
1620:
1616:
1611:
1603:
1599:
1595:
1591:
1587:
1584:
1577:
1565:
1561:
1557:
1543:
1536:
1531:
1525:
1507:
1496:
1491:
1489:
1484:
1482:
1477:
1476:
1474:
1473:
1468:
1465:
1463:
1460:
1459:
1458:
1457:
1452:
1449:
1447:
1444:
1442:
1439:
1437:
1434:
1432:
1429:
1427:
1424:
1423:
1422:
1421:
1416:
1413:
1411:
1408:
1406:
1403:
1401:
1398:
1397:
1396:
1395:
1388:
1385:
1383:
1380:
1378:
1375:
1373:
1370:
1368:
1365:
1363:
1360:
1358:
1355:
1353:
1350:
1349:
1348:
1345:
1343:
1340:
1336:
1333:
1331:
1328:
1326:
1323:
1321:
1318:
1316:
1313:
1311:
1308:
1307:
1306:
1303:
1299:
1296:
1295:
1294:
1291:
1287:
1284:
1282:
1279:
1277:
1274:
1272:
1269:
1267:
1264:
1263:
1262:
1259:
1257:
1254:
1252:
1249:
1248:
1247:
1246:
1241:
1238:
1236:
1235:Indeterminism
1233:
1229:
1226:
1225:
1224:
1221:
1217:
1214:
1213:
1212:
1209:
1208:
1207:
1206:
1202:
1198:
1197:
1194:
1191:
1190:
1187:
1183:
1182:
1162:
1138:
1135:
1131:
1126:
1120:
1112:
1108:
1098:
1094:
1077:
1069:
1066:
1063:
1060:
1054:
1048:
1042:
1033:
1029:
1012:
1002:
999:
995:
991:
988:
985:
973:
969:
952:
942:
938:
934:
931:
928:
916:
912:
908:
904:
886:
881:
878:
873:
869:
866:
860:
857:
854:
851:
848:
845:
839:
834:
830:
824:
821:
810:
806:
788:
785:
782:
777:
774:
771:
768:
765:
753:
749:
731:
728:
725:
720:
717:
714:
702:
698:
680:
677:
674:
668:
665:
662:
659:
656:
653:
644:
640:
625:
622:
616:
610:
607:
604:
575:
569:
566:
563:
548:
544:
526:
523:
497:
494:
482:
478:
463:
460:
451:
447:
443:
421:
415:
412:
409:
403:
397:
394:
386:
382:
372:
368:
351:
348:
345:
341:
335:
331:
319:
316:
299:
295:
276:
273:
270:
267:
264:
261:
258:
252:
249:
240:
236:
221:
218:
215:
212:
209:
185:
182:
179:
173:
170:
146:
143:
140:
137:
134:
131:
128:
122:
119:
110:
106:
88:
85:
82:
76:
65:
61:
53:
49:
41:
33:
19:
18722:
18710:
18676:Multivariate
18675:
18663:
18651:
18646:Wrapped Lévy
18606:
18554:Matrix gamma
18547:
18527:
18515:Normal-gamma
18508:
18474:Continuous:
18473:
18444:
18389:Tukey lambda
18376:
18368:
18363:-exponential
18360:
18352:
18343:
18334:
18325:
18319:-exponential
18316:
18260:
18227:
18194:
18156:
18143:
18070:Poly-Weibull
18015:Log-logistic
17975:
17974:Hotelling's
17906:
17748:Logit-normal
17622:Gauss–Kuzmin
17617:Flory–Schulz
17521:
17498:with finite
17385:
17376:
17352:
17346:
17334:
17329:
17310:
17304:
17279:
17275:
17269:
17257:
17252:
17238:
17219:
17213:
17196:
17171:. Retrieved
17161:
17154:
17116:
17110:
17085:. Retrieved
17075:
17068:
17053:
17028:. Retrieved
17018:
16999:
16993:
16968:
16945:. Retrieved
16942:statlect.com
16941:
16931:
16906:
16902:
16896:
16885:the original
16880:
16876:
16863:
16843:
16836:
16825:. Retrieved
16820:
16811:
16801:, retrieved
16794:the original
16773:
16769:
16756:
16740:
16727:
16715:. Retrieved
16712:itl.nist.gov
16711:
16702:
16690:. Retrieved
16686:
16676:
16666:, retrieved
16646:
16642:
16629:
16619:, retrieved
16597:
16593:
16561:
16557:
16547:
16531:
16498:
16494:
16484:
16468:
16455:
16430:
16422:
16389:
16385:
16333:
16329:
16323:
16306:
16302:
16296:
16282:(1): 13–18.
16279:
16275:
16261:94, 331-332.
16253:
16244:
16240:
16234:
16223:
16212:
16186:
16182:
16176:
16152:
16148:
16138:
16114:
16110:
16100:
16088:
16066:(1): 55–57.
16063:
16059:
16053:
16038:
16030:
16015:
16007:
15988:
15982:
15950:multifractal
15896:
15888:
15884:
15880:
15876:
15872:
15868:
15864:
15858:
15830:
15826:
15811:Methods for
15810:
15779:
15642:
15546:
15539:
15535:
15531:
15524:
15520:
15501:
15478:
15396:
15392:
15386:
15380:
15369:
15365:
15361:
15357:
15353:
15347:
15337:
15332:
15328:
15324:
15320:
15316:
15312:
15308:
15304:
15300:
15296:
15292:
15286:
15221:
15217:
15213:
15211:
15202:
15198:
15194:
15186:
15178:
15174:
15170:
15160:
15152:
15146:
15141:
15137:
15133:
15129:
15125:
15119:
14968:
14779:
14616:
14514:
14376:
14169:
14050:
13859:
13645:
13487:
13478:
13474:
13466:
13462:
13456:
13381:
13377:
13373:
13371:
13365:
13361:
13331:
13327:
13326:, Bernoulli(
13322:independent
13319:
13315:
13311:
13307:
13303:
13299:
13295:
13291:
13285:
13276:as desired.
13231:
12945:
12702:
12198:
11862:
11500:
11493:
11489:
11485:
11481:
11477:
11473:
11466:
11459:
11455:
11448:
11444:
11440:
11436:
11429:
11425:
11418:
11411:
11407:
11405:
11400:
11396:
11392:
11388:
11384:
11380:
11376:
11372:
11368:
11364:
11360:
11356:
11352:
11350:
11339:
11332:
11328:
11321:
11312:
11305:
11301:
11299:
11292:
11288:
11284:
11280:
11276:
11266:
11262:
11258:
11248:
11244:
11240:
11237:
11234:
11221:
11213:
11207:
11140:
11136:
11132:
11131:However, if
11130:
11125:
11121:
11117:
11113:
11109:
11105:
11101:
11097:
11093:
11089:
11085:
11081:
11077:
11073:
11069:
11067:
10741:
10737:
10733:
10729:
10725:
10721:
10717:
10713:
10709:
10705:
10701:
10697:
10695:
10682:
10677:
10671:
10232:
10228:
10223:
10219:
10213:
10101:
9957:
9860:
9858:
9756:
9740:
9734:
9393:
9386:
9381:
9376:
9374:
9297:
9155:
9151:
9149:
8979:
8964:
8811:
8790:
8784:
8649:
8643:
8630:
8600:
8547:
8543:
8541:
8385:
8384:/8 for even
8381:
8377:
8373:
8371:
8252:
8050:
8048:
8045:
7920:
7916:
7912:
7911:-coin and a
7908:
7900:
7896:
7892:
7890:
7785:
7780:
7776:
7772:
7768:
7764:
7760:
7756:
7752:
7748:
7746:
7631:
7626:
7622:
7544:
7478:
7474:
7472:
7428:
7299:
7295:
7255:
7218:
7211:
7108:
7100:
7004:
6947:
6888:
6801:, then only
6634:
6397:
6244:
5957:
5866:
5865:
5604:
5600:
5596:
5592:
5588:
5584:
5580:
5576:
5572:
5497:
5493:
5489:
5484:Usually the
5483:
5356:
5012:
4897:
4755:
4087:The first 6
4086:
4078:
3997:
3715:
3663:
3655:
3647:
3641:
3637:
3633:
3629:
3626:
3609:
3402:
3143:
3138:
3130:
3102:
2937:
2812:
2800:
2796:
2794:, we obtain
2736:
2727:
2717:
2713:
2706:
2702:
2699:is maximal:
2692:
2688:
2683:
2679:
2673:
2669:
2662:
2658:
2654:
2650:
2648:
2567:
2407:
2403:
2399:
2395:
2392:
2299:
2295:
2288:
2285:
2142:
2139:
2134:
2043:
2039:
2033:trials with
2025:
2021:
2007:
2004:
1997:
1908:
1904:
1901:
1750:
1746:
1742:
1738:
1728:
1688:
1651:
1632:
1618:
1614:
1609:
1601:
1563:
1553:
1541:
1534:
1467:Tree diagram
1462:Venn diagram
1426:Independence
1372:Markov chain
1314:
1256:Sample space
18760:Exponential
18609:directional
18598:Directional
18485:Generalized
18456:Multinomial
18411:continuous-
18351:Kaniadakis
18342:Kaniadakis
18333:Kaniadakis
18324:Kaniadakis
18315:Kaniadakis
18267:Tracy–Widom
18244:Skew normal
18226:Noncentral
18010:Log-Laplace
17988:Generalized
17969:Half-normal
17935:Generalized
17899:Logarithmic
17884:Exponential
17838:Chi-squared
17778:U-quadratic
17743:Kumaraswamy
17685:Continuous
17632:Logarithmic
17527:Categorical
16947:18 December
14774:paragraphs.
13356:and normal
9628:Wald method
7907:between an
7469:Tail bounds
6946:Any median
6881:is a mode.
3646:, that is,
3133:, i.e. the
2815:biased coin
1736:, we write
1680:Definitions
1586:experiments
1583:independent
1382:Random walk
1223:Determinism
1211:Probability
1155:(for fixed
18838:Categories
18755:Elliptical
18711:Degenerate
18697:Degenerate
18445:Discrete:
18404:univariate
18259:Student's
18214:Asymmetric
18193:Johnson's
18121:supported
18065:Phase-type
18020:Log-normal
18005:Log-Cauchy
17995:Kolmogorov
17913:Noncentral
17843:Noncentral
17823:Beta prime
17773:Triangular
17768:Reciprocal
17738:Irwin–Hall
17687:univariate
17667:Yule–Simon
17549:Rademacher
17491:univariate
17173:2023-10-08
17087:2023-10-08
17030:2023-10-07
16903:Biometrics
16827:2017-07-23
16803:2015-01-05
16668:2015-01-05
16621:2015-01-05
16343:2004.03280
16196:2103.17027
16189:: 109306,
16094:Proof Wiki
15975:References
15815:where the
13342:See also:
12199:Factoring
10668:Comparison
9545:the error
8807:consistent
8803:admissible
8635:consistent
8536:See also:
8376:= 1/2 and
6277:. We find
3618:Properties
2813:Suppose a
1560:statistics
1530:Galton box
1293:Experiment
1240:Randomness
1186:statistics
109:Parameters
18480:Dirichlet
18461:Dirichlet
18371:-Gaussian
18346:-Logistic
18183:Holtsmark
18155:Gaussian
18142:Fisher's
18125:real line
17627:Geometric
17607:Delaporte
17512:Bernoulli
17489:Discrete
16602:CiteSeerX
16523:121331083
16515:0013-1644
16414:189884382
16368:215238991
16360:1857-8365
16309:: 21–25.
16216:See also
16169:209923008
15756:β
15750:α
15744:
15731:−
15728:β
15717:−
15703:−
15700:α
15683:β
15677:α
15583:β
15577:α
15565:
15442:−
15419:−
15254:−
15235:σ
15090:−
15045:−
15000:−
14939:≥
14930:−
14905:−
14887:≥
14820:−
14755:−
14666:−
14653:−
14640:−
14587:−
14574:−
14561:−
14548:−
14487:−
14436:−
14349:−
14304:−
14249:−
14144:−
14115:−
14088:−
14019:−
13969:−
13949:−
13905:∈
13894:−
13874:±
13860:The rule
13828:−
13783:−
13720:∈
13709:−
13689:±
13677:σ
13671:±
13668:μ
13610:−
13597:−
13584:−
13545:−
13517:−
13430:−
13352:Binomial
13243:∼
13232:and thus
13208:−
13191:−
13126:−
13112:−
13097:−
12989:−
12975:−
12960:−
12921:−
12915:−
12901:−
12876:−
12852:−
12832:−
12814:∑
12720:−
12674:−
12660:−
12646:−
12628:−
12598:−
12587:−
12555:∑
12485:−
12471:−
12457:−
12443:−
12429:−
12406:−
12395:−
12363:∑
12238:−
12179:−
12165:−
12151:−
12137:−
12096:−
12085:−
12031:∑
11970:−
11959:−
11839:−
11825:−
11811:−
11797:−
11707:∑
11663:∣
11628:∑
11583:, by the
11553:∼
11512:∼
11300:Then log(
11179:¯
11044:−
11024:−
10949:−
10935:−
10921:−
10898:−
10865:−
10851:−
10788:∑
10762:
10576:^
10564:−
10552:^
10501:^
10434:α
10431:−
10394:−
10365:α
10334:α
10304:α
10301:−
10257:α
10218:Firstly,
10161:±
10150:^
10133:
10122:
10034:≤
10002:≠
9880:~
9816:~
9807:−
9795:~
9779:±
9773:~
9704:^
9692:−
9680:^
9661:±
9655:^
9591:α
9576:−
9553:α
9529:α
9498:α
9483:−
9415:^
9331:rule of 3
9324:^
9250:^
9211:^
9172:^
9066:^
9020:β
9001:α
8995:
8909:^
8849:β
8837:α
8831:
8764:β
8758:α
8747:α
8723:^
8686:β
8680:α
8674:
8658:conjugate
8567:^
8481:−
8456:−
8448:
8432:≥
8341:∥
8317:−
8309:
8288:≥
8222:∥
8198:−
8190:
8166:−
8133:≥
8021:−
8010:−
8001:
7989:−
7967:
7943:∥
7903:) is the
7863:∥
7839:−
7831:
7825:≤
7702:−
7684:−
7676:
7670:≤
7603:−
7585:−
7564:≥
7525:≤
7349:≤
7343:≤
7330:−
7159:−
7141:≤
7127:−
7075:−
7050:
7037:≤
7023:−
7003:A median
6988:⌉
6979:⌈
6976:≤
6970:≤
6967:⌋
6958:⌊
6869:⌋
6848:⌊
6836:⌋
6830:−
6809:⌊
6784:∉
6778:−
6702:−
6661:−
6583:⇒
6577:−
6513:⇒
6507:−
6443:⇒
6437:−
6373:−
6338:−
6155:≠
5935:−
5788:…
5776:∈
5745:−
5656:⌋
5634:⌊
5555:⌋
5552:⋅
5549:⌊
5541:, where
5529:⌋
5508:⌊
5449:
5410:
5304:
5279:≤
5231:
5214:≤
5195:
5107:−
5098:⋯
5089:−
5071:_
4982:_
4935:∑
4912:
4847:−
4809:
4774:
4718:−
4674:−
4659:−
4644:−
4614:−
4599:−
4584:−
4569:−
4554:−
4526:μ
4506:−
4488:−
4458:−
4443:−
4415:μ
4395:−
4377:−
4350:−
4322:μ
4302:−
4287:−
4259:μ
4242:−
4214:μ
4187:μ
4135:
4129:−
4115:
4100:μ
4055:−
4016:
3965:⋯
3937:
3928:⋯
3906:
3881:⋯
3862:
3844:
3805:⋯
3749:…
3680:
3547:−
3513:−
3484:−
3361:−
3347:−
3341:−
3325:−
3313:∫
3281:−
3244:−
3230:−
3203:≤
3117:⌋
3111:⌊
3080:−
3066:−
3023:⌋
3017:⌊
3002:∑
2989:≤
2918:0.059535.
2907:−
2893:−
2775:−
2763:≤
2748:−
2607:≤
2601:−
2544:−
2509:−
2369:−
2351:−
2266:−
2252:−
1971:−
1882:−
1868:−
1286:Singleton
849:π
840:
769:−
718:−
678:−
623:−
620:⌉
599:⌈
579:⌋
558:⌊
530:⌉
521:⌈
501:⌋
492:⌊
425:⌋
419:⌊
407:⌋
401:⌊
398:−
349:−
271:…
253:∈
219:−
174:∈
147:…
123:∈
18813:Category
18745:Circular
18738:Families
18723:Singular
18702:singular
18466:Negative
18413:discrete
18379:-Weibull
18337:-Weibull
18221:Logistic
18105:Discrete
18075:Rayleigh
18055:Nakagami
17978:-squared
17952:Gompertz
17801:interval
17537:Negative
17522:Binomial
17296:18698828
17182:cite web
17147:SEMATECH
17096:cite web
17062:SEMATECH
17039:cite web
16745:Archived
16473:Archived
16247:: 29–33.
16131:40233780
15907:See also
15879:) where
15843:through
15538:−
15487:variance
15475:variance
15169:since B(
15159:'s book
11387:), then
9511:quantile
8627:complete
8611:unbiased
7214:is odd).
6635:So when
6074:we find
5813:if
5754:if
5662:if
5050:are the
4000:variance
2799:= floor(
1912:, where
1596:-valued
1367:Variance
903:shannons
701:Skewness
643:Variance
67:Notation
18823:Commons
18795:Wrapped
18790:Tweedie
18785:Pearson
18780:Mixture
18687:Bingham
18586:Complex
18576:Inverse
18566:Wishart
18559:Inverse
18546:Matrix
18520:Inverse
18436:(joint)
18355:-Erlang
18209:Laplace
18100:Weibull
17957:Shifted
17940:Inverse
17925:Fréchet
17848:Inverse
17783:Uniform
17703:Arcsine
17662:Skellam
17657:Poisson
17580:support
17554:Soliton
17507:Benford
17500:support
16923:2530610
16790:2276774
16663:2685469
16406:2706397
16080:2986663
15953:measure
15855:History
15368:,
15205:, in a
15173:,
13380:,
13314:,
13006:by the
11492:,
11480:,
11443:,
11399:,
11375:,
11359:,
11338:) − 1)/
11327:) − 1)/
11124:,
11104:,
11092:,
11076:,
10740:,
10724:, then
10716:,
10704:,
9739:of 0.5/
9517:(i.e.,
9472:is the
7111:) when
5567:is the
5399:, then
5131:is the
3778:, then
3407:of the
2809:Example
2726:is the
1998:is the
1610:failure
1602:success
1598:outcome
1594:Boolean
1574:is the
1511:p = 0.5
1281:Outcome
809:Entropy
239:Support
18729:Cantor
18571:Normal
18402:Mixed
18328:-Gamma
18254:Stable
18204:Landau
18178:Gumbel
18132:Cauchy
18060:Pareto
17872:Erlang
17853:Scaled
17808:Benini
17647:Panjer
17392:
17317:
17294:
17226:
17204:
17123:
17006:
16981:
16921:
16851:
16788:
16717:18 May
16692:18 May
16661:
16604:
16521:
16513:
16443:
16412:
16404:
16366:
16358:
16167:
16129:
16078:
15995:
15784:given
15381:λ = np
15197:using
14170:Since
13490:> 5
11863:Since
11501:Since
11363:) and
11331:+ ((1/
11096:) and
10708:) and
10130:arcsin
10049:. For
9519:probit
8799:biased
7891:where
6891:median
6885:Median
6048:. For
5867:Proof:
5723:
5715:
5054:, and
5013:where
3103:where
1656:drawn
1562:, the
1548:70/256
1522:as in
1228:System
1216:Axioms
905:. For
481:Median
18451:Ewens
18277:Voigt
18249:Slash
18030:Lomax
18025:Log-t
17930:Gamma
17877:Hyper
17867:Davis
17862:Dagum
17718:Bates
17708:ARGUS
17592:Borel
17426:|X-Y|
17292:S2CID
16919:JSTOR
16888:(PDF)
16873:(PDF)
16797:(PDF)
16786:JSTOR
16766:(PDF)
16748:(PDF)
16737:(PDF)
16659:JSTOR
16639:(PDF)
16519:S2CID
16476:(PDF)
16465:(PDF)
16410:S2CID
16364:S2CID
16338:arXiv
16191:arXiv
16165:S2CID
16127:JSTOR
16076:JSTOR
15839:from
15391:: As
15376:with
15352:: As
15132:. If
11454:. If
11435:then
11424:. If
11287:) / (
11118:Z=X+Y
10734:Z=X+Y
10678:Exact
10455:0.975
10386:0.025
9513:of a
8816:as a
8656:as a
8542:When
8051:lower
7775:with
7261:When
7217:When
7105:round
5984:only
5591:and (
3612:below
2716:+ 1)
2705:+ 1)
2682:>
2672:<
2298:>
1608:) or
1514:with
1261:Event
440:(the
18700:and
18658:Kent
18085:Rice
18000:Lévy
17828:Burr
17758:PERT
17723:Beta
17672:Zeta
17564:Zipf
17481:list
17390:ISBN
17315:ISBN
17224:ISBN
17202:ISBN
17188:link
17143:NIST
17121:ISBN
17102:link
17058:NIST
17045:link
17004:ISBN
16979:ISBN
16949:2017
16849:ISBN
16719:2021
16694:2021
16511:ISSN
16441:ISBN
16402:PMID
16356:ISSN
16092:See
15993:ISBN
15887:and
15741:Beta
15562:Beta
15529:and
15071:>
15029:>
14989:and
14945:>
14893:>
14855:<
14849:<
14803:and
14738:and
14682:<
14584:>
14538:>
14497:>
14474:>
14449:>
14426:>
14388:>
14330:>
14288:>
14232:and
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14181:<
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14068:>
14030:<
13980:>
13809:>
13767:>
13629:0.3.
13626:<
13286:The
11542:and
11320:((1/
11261:~ B(
11256:and
11243:~ B(
11238:Let
11135:and
10463:1.96
10397:1.96
9984:and
9969:>
8992:Beta
8828:Beta
8805:and
8671:Beta
8648:for
8625:and
8372:For
7473:For
7427:and
7294:and
7254:and
7210:and
6717:and
6604:>
6556:<
6464:<
6416:>
6262:<
6256:<
6245:Let
6216:for
6196:and
6144:for
6109:and
5958:For
5869:Let
5619:mode
5486:mode
5480:Mode
4002:is:
3998:The
3830:and
3666:is:
2938:The
2754:<
2732:mode
2710:and
2613:<
1902:for
1726:and
1617:= 1-
1570:and
1558:and
1518:and
907:nats
547:Mode
450:Mean
18536:LKJ
17833:Chi
17424:or
17422:X-Y
17355:= 0
17284:doi
16975:130
16911:doi
16778:doi
16651:doi
16612:doi
16566:doi
16503:doi
16437:115
16394:doi
16348:doi
16311:doi
16284:doi
16201:doi
16187:182
16157:doi
16119:doi
16068:doi
16022:151
15968:XOR
15825:Pr(
15654::
15549:+ 1
15542:+ 1
15527:+ 1
15506:of
15504:PMF
15481:is
15199:x/n
15063:and
14916:and
14457:and
14322:and
14101:and
13988:and
13801:and
13372:If
11496:).
11472:is
11403:).
11351:If
11279:= (
11227:).
11122:n+m
10744:):
10738:n+m
10696:If
10113:sin
8965:(A
8795:MLE
8639:MSE
8445:exp
8380:≥ 3
8306:exp
8187:exp
7998:log
7964:log
7899:||
7828:exp
7673:exp
7144:min
7060:max
7040:min
6897:If
5301:exp
5228:log
5179::
5155:of
5151:th
4013:Var
3662:of
3627:If
2896:0.3
2878:0.3
2846:0.3
2720:- 1
2135:any
1645:of
1635:= 1
1554:In
1546:is
1544:= 4
1537:= 8
1032:PGF
915:MGF
901:in
831:log
591:or
513:or
371:CDF
298:PMF
18840::
17375:.
17290:.
17280:31
17278:.
17184:}}
17180:{{
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16332:.
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16266:^
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16064:13
16062:.
16045:52
15903:.
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15867:=
15829:=
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13650:.
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12746:Pr
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8820:,
8641:.
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7629:.
7627:np
7625:≥
7555:Pr
7516:Pr
7479:np
7477:≤
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6242:.
5844:1.
5496:,
4689:15
4662:26
4572:30
4491:12
4482:10
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3632:~
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3414::
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3141:.
2980:Pr
2805:.
2801:np
2734:.
2722:.
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2302:/2
2291:/2
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2193:Pr
1809:Pr
1771::
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1741:~
1731:∈
1701:∈
1649:.
1600::
972:CF
18548:t
18509:t
18377:q
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18361:q
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18344:κ
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15765:.
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15665:P
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15589:=
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15238:=
15222:p
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15214:n
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14260:2
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13812:9
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13680:=
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13581:1
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13421:p
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13332:p
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13320:n
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13312:n
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13264:)
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10184:)
10175:n
10170:2
10166:z
10157:)
10145:p
10137:(
10126:(
10117:2
10076:n
10073:,
10070:0
10067:=
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10058:n
10031:n
10011:n
10008:,
10005:0
9997:1
9993:n
9966:n
9938:2
9934:z
9930:+
9927:n
9920:2
9916:z
9910:2
9907:1
9902:+
9897:1
9893:n
9886:=
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9861:p
9838:2
9834:z
9830:+
9827:n
9822:)
9813:p
9804:1
9801:(
9792:p
9782:z
9770:p
9741:n
9720:.
9714:n
9710:)
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9675:p
9664:z
9650:p
9611:z
9585:2
9582:1
9573:1
9492:2
9489:1
9480:1
9460:z
9436:n
9431:1
9427:n
9421:=
9410:p
9394:n
9390:1
9387:n
9377:n
9349:.
9344:n
9341:3
9336:=
9321:p
9283:.
9277:2
9274:+
9271:n
9267:1
9262:=
9257:b
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9218:b
9208:p
9184:,
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9178:=
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9135:.
9129:1
9126:+
9123:n
9116:2
9113:1
9108:+
9105:x
9099:=
9094:s
9091:y
9088:e
9085:r
9082:f
9079:f
9076:e
9073:J
9063:p
9036:)
9031:2
9028:1
9023:=
9017:,
9012:2
9009:1
9004:=
8998:(
8950:.
8944:2
8941:+
8938:n
8933:1
8930:+
8927:x
8921:=
8916:b
8906:p
8879:)
8876:1
8873:,
8870:0
8867:(
8864:U
8861:=
8858:)
8855:1
8852:=
8846:,
8843:1
8840:=
8834:(
8791:n
8770:.
8761:+
8755:+
8752:n
8744:+
8741:x
8735:=
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8689:)
8683:,
8677:(
8650:p
8631:x
8586:.
8581:n
8578:x
8573:=
8564:p
8548:p
8544:n
8510:.
8506:)
8500:2
8495:)
8489:n
8486:k
8476:2
8473:1
8467:(
8462:n
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8437:1
8429:)
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8411:n
8408:;
8405:k
8402:(
8399:F
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8348:)
8344:p
8336:n
8333:k
8327:(
8323:D
8320:n
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8282:p
8279:,
8276:n
8273:;
8270:k
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8264:F
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8234:)
8229:)
8225:p
8217:n
8214:k
8208:(
8204:D
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8154:n
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8112:(
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8064:(
8061:F
8030:.
8024:p
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8007:1
7995:)
7992:a
7986:1
7983:(
7980:+
7975:p
7972:a
7961:)
7958:a
7955:(
7952:=
7949:)
7946:p
7940:a
7937:(
7934:D
7921:p
7917:a
7913:p
7909:a
7901:p
7897:a
7895:(
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7875:)
7870:)
7866:p
7858:n
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7849:(
7845:D
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7813:n
7810:;
7807:k
7804:(
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7781:n
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7773:n
7769:k
7765:p
7763:,
7761:n
7759:;
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7755:(
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7731:,
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7667:)
7664:p
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7623:k
7609:)
7606:p
7600:1
7597:,
7594:n
7591:;
7588:k
7582:n
7579:(
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7573:=
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7567:k
7561:X
7558:(
7545:k
7531:)
7528:k
7522:X
7519:(
7513:=
7510:)
7507:p
7504:,
7501:n
7498:;
7495:k
7492:(
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7475:k
7450:2
7447:n
7442:=
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7429:n
7413:2
7410:1
7405:=
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7380:)
7375:1
7372:+
7369:n
7364:(
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7346:m
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7322:(
7315:2
7312:1
7300:m
7296:n
7280:2
7277:1
7272:=
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7240:2
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7232:=
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7219:p
7212:n
7196:2
7193:1
7188:=
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7165:}
7162:p
7156:1
7153:,
7150:p
7147:{
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7124:m
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7107:(
7101:m
7096:.
7084:}
7081:}
7078:p
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7066:p
7063:{
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7053:2
7043:{
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7016:|
7005:m
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6961:n
6948:m
6943:.
6931:p
6928:n
6908:p
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6866:p
6863:)
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6857:+
6854:n
6851:(
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6839:+
6833:1
6827:p
6824:)
6821:1
6818:+
6815:n
6812:(
6788:Z
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6699:p
6696:)
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6687:n
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6616:)
6613:k
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6589:(
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6571:)
6568:1
6565:+
6562:n
6559:(
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6546:)
6543:k
6540:(
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6534:=
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6528:1
6525:+
6522:k
6519:(
6516:f
6510:1
6504:p
6501:)
6498:1
6495:+
6492:n
6489:(
6486:=
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6476:)
6473:k
6470:(
6467:f
6461:)
6458:1
6455:+
6452:k
6449:(
6446:f
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6434:p
6431:)
6428:1
6425:+
6422:n
6419:(
6413:k
6394:.
6379:)
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6370:1
6367:(
6364:)
6361:1
6358:+
6355:k
6352:(
6347:p
6344:)
6341:k
6335:n
6332:(
6326:=
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6317:k
6314:(
6311:f
6306:)
6303:1
6300:+
6297:k
6294:(
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6265:1
6259:p
6253:0
6230:1
6227:=
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6132:0
6129:=
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6123:k
6120:(
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6097:1
6094:=
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6088:n
6085:(
6082:f
6062:1
6059:=
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6036:1
6033:=
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6027:0
6024:(
6021:f
6001:)
5998:0
5995:(
5992:f
5972:0
5969:=
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5943:.
5938:k
5932:n
5928:q
5922:k
5918:p
5911:)
5906:k
5903:n
5898:(
5892:=
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5886:k
5883:(
5880:f
5841:+
5838:n
5835:=
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5829:)
5826:1
5823:+
5820:n
5817:(
5807:n
5800:,
5797:}
5794:n
5791:,
5785:,
5782:1
5779:{
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5767:1
5764:+
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5758:(
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5732:+
5729:n
5726:(
5712:p
5708:)
5705:1
5702:+
5699:n
5696:(
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5675:1
5672:+
5669:n
5666:(
5653:p
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5646:1
5643:+
5640:n
5637:(
5628:{
5623:=
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5601:p
5597:p
5593:n
5589:p
5585:n
5581:p
5577:p
5573:n
5526:p
5523:)
5520:1
5517:+
5514:n
5511:(
5498:p
5494:n
5492:(
5490:B
5463:c
5459:]
5455:X
5452:[
5446:E
5426:]
5421:c
5417:X
5413:[
5407:E
5387:)
5382:p
5379:n
5374:(
5371:O
5368:=
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5340:.
5336:)
5330:p
5327:n
5324:2
5318:2
5314:c
5308:(
5296:c
5292:)
5288:p
5285:n
5282:(
5274:c
5269:)
5263:)
5260:1
5257:+
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5251:p
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5245:(
5241:/
5237:c
5234:(
5224:c
5219:(
5211:]
5206:c
5202:X
5198:[
5192:E
5163:n
5139:k
5119:)
5116:1
5113:+
5110:k
5104:n
5101:(
5095:)
5092:1
5086:n
5083:(
5080:n
5077:=
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5063:n
5036:}
5031:k
5028:c
5023:{
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4993:k
4989:p
4979:k
4974:n
4969:}
4964:k
4961:c
4956:{
4950:c
4945:0
4942:=
4939:k
4931:=
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4923:c
4919:X
4915:[
4909:E
4879:,
4874:2
4870:p
4864:2
4860:n
4856:+
4853:)
4850:p
4844:1
4841:(
4838:p
4835:n
4832:=
4825:]
4820:2
4816:X
4812:[
4806:E
4799:,
4796:p
4793:n
4790:=
4783:]
4780:X
4777:[
4771:E
4737:.
4734:)
4729:2
4725:)
4721:p
4715:1
4712:(
4707:2
4703:p
4697:2
4693:n
4686:+
4683:)
4680:)
4677:p
4671:1
4668:(
4665:p
4656:5
4653:(
4650:)
4647:p
4641:1
4638:(
4635:p
4632:n
4629:5
4626:+
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4620:)
4617:p
4611:1
4608:(
4605:p
4602:4
4596:1
4593:(
4590:)
4587:p
4581:1
4578:(
4575:p
4566:1
4563:(
4560:)
4557:p
4551:1
4548:(
4545:p
4542:n
4539:=
4530:6
4518:,
4515:)
4512:)
4509:p
4503:1
4500:(
4497:p
4494:)
4485:n
4479:(
4476:+
4473:1
4470:(
4467:)
4464:p
4461:2
4455:1
4452:(
4449:)
4446:p
4440:1
4437:(
4434:p
4431:n
4428:=
4419:5
4407:,
4404:)
4401:)
4398:p
4392:1
4389:(
4386:p
4383:)
4380:6
4374:n
4371:3
4368:(
4365:+
4362:1
4359:(
4356:)
4353:p
4347:1
4344:(
4341:p
4338:n
4335:=
4326:4
4314:,
4311:)
4308:p
4305:2
4299:1
4296:(
4293:)
4290:p
4284:1
4281:(
4278:p
4275:n
4272:=
4263:3
4251:,
4248:)
4245:p
4239:1
4236:(
4233:p
4230:n
4227:=
4218:2
4206:,
4203:0
4200:=
4191:1
4158:]
4152:c
4148:)
4144:]
4141:X
4138:[
4132:E
4126:X
4123:(
4119:[
4112:E
4109:=
4104:c
4064:.
4061:)
4058:p
4052:1
4049:(
4046:p
4043:n
4040:=
4037:q
4034:p
4031:n
4028:=
4025:)
4022:X
4019:(
3983:.
3980:p
3977:n
3974:=
3971:p
3968:+
3962:+
3959:p
3956:=
3953:]
3948:n
3944:X
3940:[
3934:E
3931:+
3925:+
3922:]
3917:1
3913:X
3909:[
3903:E
3900:=
3897:]
3892:n
3888:X
3884:+
3878:+
3873:1
3869:X
3865:[
3859:E
3856:=
3853:]
3850:X
3847:[
3841:E
3816:n
3812:X
3808:+
3802:+
3797:1
3793:X
3789:=
3786:X
3776:p
3760:n
3756:X
3752:,
3746:,
3741:1
3737:X
3726:p
3722:n
3718:X
3701:.
3698:p
3695:n
3692:=
3689:]
3686:X
3683:[
3677:E
3664:X
3656:p
3652:n
3648:X
3644:)
3642:p
3638:n
3636:(
3634:B
3630:X
3595:.
3591:)
3587:)
3584:1
3581:+
3578:k
3575:(
3572:2
3569:=
3564:2
3560:d
3556:,
3553:)
3550:k
3544:n
3541:(
3538:2
3535:=
3530:1
3526:d
3522:;
3516:k
3510:n
3505:1
3502:+
3499:k
3491:p
3487:p
3481:1
3475:=
3472:x
3468:(
3457:F
3453:F
3449:=
3446:)
3443:p
3440:,
3437:n
3434:;
3431:k
3428:(
3425:F
3410:F
3384:.
3381:t
3378:d
3372:k
3368:)
3364:t
3358:1
3355:(
3350:1
3344:k
3338:n
3334:t
3328:p
3322:1
3317:0
3306:)
3301:k
3298:n
3293:(
3287:)
3284:k
3278:n
3275:(
3272:=
3262:)
3259:1
3256:+
3253:k
3250:,
3247:k
3241:n
3238:(
3233:p
3227:1
3223:I
3219:=
3209:)
3206:k
3200:X
3197:(
3191:=
3184:)
3181:p
3178:,
3175:n
3172:;
3169:k
3166:(
3163:F
3139:k
3131:k
3114:k
3088:,
3083:i
3077:n
3073:)
3069:p
3063:1
3060:(
3055:i
3051:p
3044:)
3039:i
3036:n
3031:(
3020:k
3012:0
3009:=
3006:i
2998:=
2995:)
2992:k
2986:X
2983:(
2977:=
2974:)
2971:p
2968:,
2965:n
2962:;
2959:k
2956:(
2953:F
2915:=
2910:4
2904:6
2900:)
2890:1
2887:(
2882:4
2871:)
2866:4
2863:6
2858:(
2852:=
2849:)
2843:,
2840:6
2837:,
2834:4
2831:(
2828:f
2803:)
2797:M
2778:p
2772:1
2769:+
2766:M
2760:p
2757:n
2751:p
2745:M
2724:M
2718:p
2714:n
2712:(
2707:p
2703:n
2701:(
2697:f
2693:p
2689:n
2684:M
2680:k
2674:M
2670:k
2665:)
2663:p
2659:n
2655:k
2653:(
2651:f
2634:.
2631:p
2628:)
2625:1
2622:+
2619:n
2616:(
2610:M
2604:1
2598:p
2595:)
2592:1
2589:+
2586:n
2583:(
2570:M
2550:)
2547:p
2541:1
2538:(
2535:)
2532:1
2529:+
2526:k
2523:(
2518:p
2515:)
2512:k
2506:n
2503:(
2497:=
2491:)
2488:p
2485:,
2482:n
2479:,
2476:k
2473:(
2470:f
2465:)
2462:p
2459:,
2456:n
2453:,
2450:1
2447:+
2444:k
2441:(
2438:f
2422:k
2418:k
2414:k
2410:)
2408:p
2404:n
2400:k
2398:(
2396:f
2378:.
2375:)
2372:p
2366:1
2363:,
2360:n
2357:,
2354:k
2348:n
2345:(
2342:f
2339:=
2336:)
2333:p
2330:,
2327:n
2324:,
2321:k
2318:(
2315:f
2300:n
2296:k
2289:n
2269:k
2263:n
2259:)
2255:p
2249:1
2246:(
2241:k
2237:p
2230:)
2225:k
2222:n
2217:(
2211:=
2208:)
2205:k
2202:=
2199:X
2196:(
2170:)
2165:k
2162:n
2157:(
2143:q
2140:p
2131:n
2127:k
2110:)
2105:k
2102:n
2097:(
2071:)
2066:k
2063:n
2058:(
2044:k
2042:-
2040:n
2035:k
2031:n
2026:k
2024:-
2022:n
2017:k
2013:n
2008:q
2005:p
1980:!
1977:)
1974:k
1968:n
1965:(
1962:!
1959:k
1954:!
1951:n
1945:=
1939:)
1934:k
1931:n
1926:(
1909:n
1905:k
1885:k
1879:n
1875:)
1871:p
1865:1
1862:(
1857:k
1853:p
1846:)
1841:k
1838:n
1833:(
1827:=
1824:)
1821:k
1818:=
1815:X
1812:(
1806:=
1803:)
1800:p
1797:,
1794:n
1791:,
1788:k
1785:(
1782:f
1765:p
1761:n
1757:k
1753:)
1751:p
1747:n
1745:(
1743:B
1739:X
1729:p
1711:N
1699:n
1694:X
1674:n
1670:N
1662:N
1654:n
1633:n
1619:p
1615:q
1606:p
1580:n
1572:p
1568:n
1550:.
1542:k
1535:n
1520:k
1516:n
1494:e
1487:t
1480:v
1175:)
1163:n
1139:q
1136:p
1132:n
1127:=
1124:)
1121:p
1118:(
1113:n
1109:g
1078:n
1074:]
1070:z
1067:p
1064:+
1061:q
1058:[
1055:=
1052:)
1049:z
1046:(
1043:G
1013:n
1009:)
1003:t
1000:i
996:e
992:p
989:+
986:q
983:(
953:n
949:)
943:t
939:e
935:p
932:+
929:q
926:(
887:)
882:n
879:1
874:(
870:O
867:+
864:)
861:q
858:p
855:n
852:e
846:2
843:(
835:2
825:2
822:1
789:q
786:p
783:n
778:q
775:p
772:6
766:1
732:q
729:p
726:n
721:p
715:q
684:)
681:p
675:1
672:(
669:p
666:n
663:=
660:q
657:p
654:n
626:1
617:p
614:)
611:1
608:+
605:n
602:(
576:p
573:)
570:1
567:+
564:n
561:(
527:p
524:n
498:p
495:n
464:p
461:n
444:)
428:)
422:k
416:+
413:1
410:,
404:k
395:n
392:(
387:q
383:I
352:k
346:n
342:q
336:k
332:p
325:)
320:k
317:n
312:(
280:}
277:n
274:,
268:,
265:1
262:,
259:0
256:{
250:k
222:p
216:1
213:=
210:q
189:]
186:1
183:,
180:0
177:[
171:p
150:}
144:,
141:2
138:,
135:1
132:,
129:0
126:{
120:n
92:)
89:p
86:,
83:n
80:(
77:B
34:.
20:)
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