12993:
11733:
12988:{\displaystyle {\begin{aligned}f\left(x;{\tfrac {4}{3}},0,1,0\right)&={\frac {3^{\frac {5}{4}}}{4{\sqrt {2\pi }}}}{\frac {\Gamma \left({\tfrac {7}{12}}\right)\Gamma \left({\tfrac {11}{12}}\right)}{\Gamma \left({\tfrac {6}{12}}\right)\Gamma \left({\tfrac {8}{12}}\right)}}{}_{2}F_{2}\left({\tfrac {7}{12}},{\tfrac {11}{12}};{\tfrac {6}{12}},{\tfrac {8}{12}};{\tfrac {3^{3}x^{4}}{4^{4}}}\right)-{\frac {3^{\frac {11}{4}}x^{3}}{4^{3}{\sqrt {2\pi }}}}{\frac {\Gamma \left({\tfrac {13}{12}}\right)\Gamma \left({\tfrac {17}{12}}\right)}{\Gamma \left({\tfrac {18}{12}}\right)\Gamma \left({\tfrac {15}{12}}\right)}}{}_{2}F_{2}\left({\tfrac {13}{12}},{\tfrac {17}{12}};{\tfrac {18}{12}},{\tfrac {15}{12}};{\tfrac {3^{3}x^{4}}{4^{4}}}\right)\\f\left(x;{\tfrac {3}{2}},0,1,0\right)&={\frac {\Gamma \left({\tfrac {5}{3}}\right)}{\pi }}{}_{2}F_{3}\left({\tfrac {5}{12}},{\tfrac {11}{12}};{\tfrac {1}{3}},{\tfrac {1}{2}},{\tfrac {5}{6}};-{\tfrac {2^{2}x^{6}}{3^{6}}}\right)-{\frac {x^{2}}{3\pi }}{}_{3}F_{4}\left({\tfrac {3}{4}},1,{\tfrac {5}{4}};{\tfrac {2}{3}},{\tfrac {5}{6}},{\tfrac {7}{6}},{\tfrac {4}{3}};-{\tfrac {2^{2}x^{6}}{3^{6}}}\right)+{\frac {7x^{4}\Gamma \left({\tfrac {4}{3}}\right)}{3^{4}\pi ^{2}}}{}_{2}F_{3}\left({\tfrac {13}{12}},{\tfrac {19}{12}};{\tfrac {7}{6}},{\tfrac {3}{2}},{\tfrac {5}{3}};-{\tfrac {2^{2}x^{6}}{3^{6}}}\right)\end{aligned}}}
13840:
13051:
13835:{\displaystyle {\begin{aligned}f\left(x;{\tfrac {2}{3}},0,1,0\right)&={\frac {\sqrt {3}}{6{\sqrt {\pi }}|x|}}\exp \left({\tfrac {2}{27}}x^{-2}\right)W_{-{\frac {1}{2}},{\frac {1}{6}}}\left({\tfrac {4}{27}}x^{-2}\right)\\f\left(x;{\tfrac {2}{3}},1,1,0\right)&={\frac {\sqrt {3}}{{\sqrt {\pi }}|x|}}\exp \left(-{\tfrac {16}{27}}x^{-2}\right)W_{{\frac {1}{2}},{\frac {1}{6}}}\left({\tfrac {32}{27}}x^{-2}\right)\\f\left(x;{\tfrac {3}{2}},1,1,0\right)&={\begin{cases}{\frac {\sqrt {3}}{{\sqrt {\pi }}|x|}}\exp \left({\frac {1}{27}}x^{3}\right)W_{{\frac {1}{2}},{\frac {1}{6}}}\left(-{\frac {2}{27}}x^{3}\right)&x<0\\{}\\{\frac {\sqrt {3}}{6{\sqrt {\pi }}|x|}}\exp \left({\frac {1}{27}}x^{3}\right)W_{-{\frac {1}{2}},{\frac {1}{6}}}\left({\frac {2}{27}}x^{3}\right)&x\geq 0\end{cases}}\end{aligned}}}
83:
113:
67:
4658:
37:
7504:
6439:
7427:
16549:
4234:
16559:
6179:
7223:
11056:
9233:
4653:{\displaystyle {\begin{aligned}L_{\alpha }(x)&={\frac {1}{\pi }}\Re \left\\&={\frac {2}{\pi }}\int _{0}^{\infty }e^{-\operatorname {Re} (q)\,t^{\alpha }}\sin(tx)\sin(-\operatorname {Im} (q)\,t^{\alpha })\,dt,{\text{ or }}\\&={\frac {2}{\pi }}\int _{0}^{\infty }e^{-{\text{Re}}(q)\,t^{\alpha }}\cos(tx)\cos(\operatorname {Im} (q)\,t^{\alpha })\,dt.\end{aligned}}}
11686:
5255:
5780:
5063:
9703:
6954:
6993:
6434:{\displaystyle \int _{0}^{\infty }{\frac {1}{\nu }}\left({\frac {1}{2}}e^{-{\frac {|z|}{\nu }}}\right)\left({\frac {\alpha }{\Gamma ({\frac {1}{\alpha }})}}{\frac {1}{\nu }}L_{\alpha }\left({\frac {1}{\nu }}\right)\right)\,d\nu ={\frac {1}{2}}{\frac {\alpha }{\Gamma ({\frac {1}{\alpha }})}}e^{-|z|^{\alpha }},{\text{ where }}\alpha <1.}
2776:
10825:
10023:
3690:
11447:
9884:
8904:
11494:
3015:
1238:
7781:
of
Gaussian random variables (all with mean zero), with the variance being drawn from a standard Lévy distribution. And in fact this is a special case of a more general theorem (See p. 59 of ) which allows any symmetric alpha-stable distribution to be viewed in this way (with the alpha parameter
10515:
BATSE hard x-ray solar flares in
December 2001. Analysis of the Lévy statistical signature revealed that two different memory signatures were evident; one related to the solar cycle and the second whose origin appears to be associated with a localized or combination of localized solar active region
2601:
5078:
9358:
itself, so the inversion method cannot be used to generate stable-distributed variates. Other standard approaches like the rejection method would require tedious computations. An elegant and efficient solution was proposed by
Chambers, Mallows and Stuck (CMS), who noticed that a certain integral
5567:
9495:
6697:
2222:
The reason this gives a stable distribution is that the characteristic function for the sum of two independent random variables equals the product of the two corresponding characteristic functions. Adding two random variables from a stable distribution gives something with the same values of
2127:
13971:
which has methods of generation, fitting, probability density, cumulative distribution function, characteristic and moment generating functions, quantile and related functions, convolution and affine transformations of stable distributions. It uses modernised algorithms improved by John P.
4926:
2606:
1986:
10464:
of the stable family. It was the seeming departure from normality along with the demand for a self-similar model for financial data (i.e. the shape of the distribution for yearly asset price changes should resemble that of the constituent daily or monthly price changes) that led
9889:
3522:
2426:
The parametrization of stable distributions is not unique. Nolan tabulates 11 parametrizations seen in the literature and gives conversion formulas. The two most commonly used parametrizations are the one above (Nolan's "1") and the one immediately below (Nolan's "0").
13912:. It calculates the density (pdf), cumulative distribution function (cdf) and quantiles for a general stable distribution, and performs maximum likelihood estimation of stable parameters and some exploratory data analysis techniques for assessing the fit of a data set.
11125:
9736:
10333:
1452:
the properly normed sum of a set of random variables, each with finite variance, will tend toward a normal distribution as the number of variables increases. Without the finite variance assumption, the limit may be a stable distribution that is not normal.
4027:
7218:{\displaystyle {\begin{aligned}\mu &=\mu _{1}+\mu _{2}\\c&=\left(c_{1}^{\alpha }+c_{2}^{\alpha }\right)^{\frac {1}{\alpha }}\\\beta &={\frac {\beta _{1}c_{1}^{\alpha }+\beta _{2}c_{2}^{\alpha }}{c_{1}^{\alpha }+c_{2}^{\alpha }}}\end{aligned}}}
6126:
11051:{\displaystyle f\left(x;{\tfrac {1}{3}},0,1,0\right)=\Re \left({\frac {2e^{-{\frac {i\pi }{4}}}}{3{\sqrt {3}}\pi }}{\frac {1}{\sqrt {x^{3}}}}S_{0,{\frac {1}{3}}}\left({\frac {2e^{\frac {i\pi }{4}}}{3{\sqrt {3}}}}{\frac {1}{\sqrt {x}}}\right)\right)}
2895:
2459:
2007:
9228:{\displaystyle {\begin{aligned}L_{\alpha }(x)&={\frac {1}{\pi }}\Re \left\\&={\frac {1}{\pi }}\sum _{n=1}^{\infty }{\frac {-\sin(n(\alpha +1)\pi )}{n!}}\left({\frac {1}{x}}\right)^{\alpha n+1}\Gamma (\alpha n+1)\end{aligned}}}
1112:
798:
11681:{\displaystyle f\left(x;{\tfrac {1}{3}},1,1,0\right)={\frac {1}{\pi }}{\frac {2{\sqrt {2}}}{3^{\frac {7}{4}}}}{\frac {1}{\sqrt {x^{3}}}}K_{\frac {1}{3}}\left({\frac {4{\sqrt {2}}}{3^{\frac {9}{4}}}}{\frac {1}{\sqrt {x}}}\right)}
5250:{\displaystyle {\mathfrak {N}}_{\alpha }(\nu ;\nu _{0},\theta )={\frac {\alpha }{\Gamma ({\frac {1}{\alpha }})}}{\frac {1}{\nu -\nu _{0}}}L_{\alpha }\left({\frac {\theta }{\nu -\nu _{0}}}\right),{\text{ where }}\nu >\nu _{0}}
3489:
5775:{\displaystyle {\mathfrak {N}}_{\frac {1}{2}}(\nu ;\nu _{0},\theta )={\frac {1}{4{\sqrt {\pi }}\theta ^{3/2}}}(\nu -\nu _{0})^{1/2}e^{-{\frac {\nu -\nu _{0}}{4\theta }}},{\text{ where }}\nu >\nu _{0},\qquad \theta >0.}
3171:
1844:
5058:{\displaystyle {\mathfrak {N}}_{\alpha }(\nu )={\frac {\alpha }{\Gamma \left({\frac {1}{\alpha }}\right)}}{\frac {1}{\nu }}L_{\alpha }\left({\frac {1}{\nu }}\right),{\text{ where }}\nu >0{\text{ and }}\alpha <1.}
2278:
is real and goes from 0 to 1 without decreasing), but the characteristic functions given above will be legitimate so long as the parameters are in their ranges. The value of the characteristic function at some value
4845:
8324:
1833:
8773:
8523:
5532:
4191:
9438:
10443:
random variables. While other approaches have been proposed in the literature, including application of
Bergström and LePage series expansions, the CMS method is regarded as the fastest and the most accurate.
10222:
5914:
13056:
6444:
This is called the "lambda decomposition" (See
Section 4 of ) since the right hand side was named as "symmetric lambda distribution" in Lihn's former works. However, it has several more popular names such as
9698:{\displaystyle X=\left(1+\zeta ^{2}\right)^{\frac {1}{2\alpha }}{\frac {\sin(\alpha (U+\xi ))}{(\cos(U))^{\frac {1}{\alpha }}}}\left({\frac {\cos(U-\alpha (U+\xi ))}{W}}\right)^{\frac {1-\alpha }{\alpha }},}
6949:{\displaystyle \exp \left(it\mu _{1}+it\mu _{2}-|c_{1}t|^{\alpha }-|c_{2}t|^{\alpha }+i\beta _{1}|c_{1}t|^{\alpha }\operatorname {sgn}(t)\Phi +i\beta _{2}|c_{2}t|^{\alpha }\operatorname {sgn}(t)\Phi \right)}
6998:
927:
11738:
5298:
3396:
681:
13886:
3180:
A stable distribution is therefore specified by the above four parameters. It can be shown that any non-degenerate stable distribution has a smooth (infinitely differentiable) density function. If
2771:{\displaystyle \Phi ={\begin{cases}\left(|\gamma t|^{1-\alpha }-1\right)\tan \left({\tfrac {\pi \alpha }{2}}\right)&\alpha \neq 1\\-{\frac {2}{\pi }}\log |\gamma t|&\alpha =1\end{cases}}}
8909:
4239:
7813:. Several closed form expressions having rather simple expressions in terms of special functions are available. In the table below, PDFs expressible by elementary functions are indicated by an
3319:
1837:
Although the probability density function for a general stable distribution cannot be written analytically, the general characteristic function can be expressed analytically. A random variable
6516:
3058:
851:
10217:
10080:
8577:
10710:
4229:
10575:
10381:
10018:{\displaystyle \zeta =-\beta \tan {\frac {\pi \alpha }{2}},\qquad \xi ={\begin{cases}{\frac {1}{\alpha }}\arctan(-\zeta )&\alpha \neq 1\\{\frac {\pi }{2}}&\alpha =1\end{cases}}}
3685:{\displaystyle f(x)\sim {\frac {1}{|x|^{1+\alpha }}}\left(c^{\alpha }(1+\operatorname {sgn}(x)\beta )\sin \left({\frac {\pi \alpha }{2}}\right){\frac {\Gamma (\alpha +1)}{\pi }}\right)}
3231:
980:
11442:{\displaystyle f\left(x;{\tfrac {1}{2}},0,1,0\right)={\frac {1}{\sqrt {2\pi |x|^{3}}}}\left(\sin \left({\tfrac {1}{4|x|}}\right)\left+\cos \left({\tfrac {1}{4|x|}}\right)\left\right)}
3912:
8863:
7809:. Fox H-Functions can also be used to express the stable probability density functions. For simple rational numbers, the closed form expression is often in terms of less complicated
4111:
2890:
9879:{\displaystyle X={\frac {1}{\xi }}\left\{\left({\frac {\pi }{2}}+\beta U\right)\tan U-\beta \log \left({\frac {{\frac {\pi }{2}}W\cos U}{{\frac {\pi }{2}}+\beta U}}\right)\right\},}
4786:
9280:
6015:
82:
5846:
5423:
4737:
5340:
1344:
5819:
1491:
10816:
5464:
165:
4886:
1105:
8899:
6587:
13042:
10772:
10639:
9490:
5947:
4053:
3720:
1438:
14541:
7544:
7471:
6473:
3866:
3783:
3749:
3517:
3104:
2359:
2209:
1408:
408:
312:
251:
11724:
8070:
7733:
6674:. Since convolution is equivalent to multiplication of the Fourier-transformed function, it follows that the product of two stable characteristic functions with the same
5973:
3401:
1712:
16588:
2175:
10407:
9731:
9327:
7999:
7658:
7593:
7497:
6641:
3812:
2454:
1374:
707:
598:
562:
526:
341:
11485:
10433:
8096:
8025:
7759:
7684:
6163:
6004:
5559:
5367:
3892:
2385:
2314:
486:
447:
277:
10487:
10103:
7848:
7803:
6692:
6672:
5387:
3078:
2798:
2416:
2241:
1677:
1312:
106:
60:
10143:
7898:
7629:
6988:
6551:
2818:
2261:
1657:
186:
11116:
11087:
10163:
9356:
3010:{\displaystyle y={\begin{cases}{\frac {x-\mu }{\gamma }}&\alpha \neq 1\\{\frac {x-\mu }{\gamma }}-\beta {\frac {2}{\pi }}\ln \gamma &\alpha =1\end{cases}}}
1745:
1233:{\displaystyle \Phi ={\begin{cases}\tan {\tfrac {\pi \alpha }{2}}&{\text{if }}\alpha \neq 1\\-{\tfrac {2}{\pi }}\log |t|&{\text{if }}\alpha =1\end{cases}}}
13927:, which includes among the Gaussian and Cauchy distributions also an implementation of the Levy alpha-stable distribution, both with and without a skew parameter.
7782:
of the mixture distribution equal to twice the alpha parameter of the mixing distribution—and the beta parameter of the mixing distribution always equal to one).
15124:
Zlotarev, V. M. (1961). "Expression of the density of a stable distribution with exponent alpha greater than one by means of a frequency with exponent 1/alpha".
13937:
implementation for the Stable distribution pdf, cdf, random number, quantile and fitting functions (along with a benchmark replication package and an R package).
10123:
9461:
9381:
4680:
9250:, a general method which relies on the quantiles was developed by McCulloch and works for both symmetric and skew stable distributions and stability parameter
4912:
14883:
8147:
2596:{\displaystyle \varphi (t;\alpha ,\beta ,\gamma ,\delta )=\exp \left(it\delta -|\gamma t|^{\alpha }\left(1-i\beta \operatorname {sgn}(t)\Phi \right)\right)}
1750:
713:
8582:
8335:
15207:
3113:
2122:{\displaystyle \Phi ={\begin{cases}\tan \left({\frac {\pi \alpha }{2}}\right)&\alpha \neq 1\\-{\frac {2}{\pi }}\log |t|&\alpha =1\end{cases}}}
14952:
Janicki, Aleksander; Kokoszka, Piotr (1992). "Computer investigation of the Rate of
Convergence of Lepage Type Series to α-Stable Random Variables".
15126:
Selected
Translations in Mathematical Statistics and Probability (Translated from the Russian Article: Dokl. Akad. Nauk SSSR. 98, 735–738 (1954))
6009:
Another approach to derive the stable count distribution is to use the
Laplace transform of the one-sided stable distribution, (Section 2.4 of )
8901:. There is no real part to sum. Instead, the integral of the characteristic function should be carried out on the negative axis, which yields:
4795:
7562:
5473:
4116:
1683:
989:
1981:{\displaystyle \varphi (t;\alpha ,\beta ,c,\mu )=\exp \left(it\mu -|ct|^{\alpha }\left(1-i\beta \operatorname {sgn}(t)\Phi \right)\right)}
15336:
13947:
by
Diethelm Wuertz, Martin Maechler and Rmetrics core team members. Computes stable density, probability, quantiles, and random numbers.
9386:
3248:
16562:
15819:
7777:
Note that the above three distributions are also connected, in the following way: A standard Cauchy random variable can be viewed as a
5855:
15727:
14207:
2430:
The parametrization above is easiest to use for theoretical work, but its probability density is not continuous in the parameters at
3022:
857:
16514:
1493:
as "Pareto–Lévy distributions", which he regarded as better descriptions of stock and commodity prices than normal distributions.
16380:
15592:
15351:
15200:
15066:
Hopcraft, K. I.; Jakeman, E.; Tanner, R. M. J. (1999). "Lévy random walks with fluctuating step number and multiscale behavior".
2839:
In either parametrization one can make a linear transformation of the random variable to get a random variable whose density is
16275:
16039:
112:
7281:
is α-stable for some 0 < α ≤ 2 if and only if there is an independent, identically distributed sequence of random variables
5260:
3328:
15713:
14998:
14821:
14783:
14325:
14285:
9441:
622:
14390:
Penson, K. A.; Górska, K. (2010-11-17). "Exact and
Explicit Probability Densities for One-Sided Lévy Stable Distributions".
16034:
15978:
15876:
15638:
15276:
10328:{\displaystyle Y={\begin{cases}cX+\mu &\alpha \neq 1\\cX+{\frac {2}{\pi }}\beta c\log c+\mu &\alpha =1\end{cases}}}
7255:, and others) over the period from 1920 to 1937. The first published complete proof (in French) of the GCLT was in 1937 by
14909:
Mantegna, Rosario Nunzio (1994). "Fast, accurate algorithm for numerical simulation of Lévy stable stochastic processes".
10524:
A number of cases of analytically expressible stable distributions are known. Let the stable distribution be expressed by
8799:
has therefore been dropped.) Expressing the first exponential as a series will yield another series in positive powers of
7378:
Here → means the sequence of random variable sums converges in distribution; i.e., the corresponding distributions satisfy
16320:
16054:
15907:
15582:
15326:
7227:
In each case, it can be shown that the resulting parameters lie within the required intervals for a stable distribution.
15784:
66:
16552:
16224:
16200:
15779:
15193:
16583:
16421:
16298:
16259:
16231:
16205:
16123:
16049:
15472:
15220:
14893:
14551:
7778:
6599:
6481:
1262:
13954:
803:
16603:
16409:
16375:
16241:
16236:
16081:
15889:
15587:
15341:
14160:
13891:
10168:
10031:
8528:
2275:
366:
9247:
7410:
In other words, if sums of independent, identically distributed random variables converge in distribution to some
16159:
16072:
16044:
15953:
15902:
15774:
15557:
15522:
10650:
6694:
will yield another such characteristic function. The product of two stable characteristic functions is given by:
4196:
4022:{\displaystyle L_{\alpha }(x)=f\left(x;\alpha ,1,\cos \left({\frac {\alpha \pi }{2}}\right)^{1/\alpha },0\right)}
15109:
Uchaikin, V. V.; Zolotarev, V. M. (1999). "Chance And Stability – Stable Distributions And Their Applications".
10527:
10338:
3183:
932:
16173:
16090:
15927:
15851:
15674:
15552:
15527:
15391:
15386:
15381:
15013:
6446:
2317:
2274:
Not every function is the characteristic function of a legitimate probability distribution (that is, one whose
36:
16598:
16489:
16355:
16063:
15912:
15844:
15829:
15722:
15696:
15628:
15467:
15361:
15356:
15298:
15283:
13950:
8813:
6121:{\displaystyle \int _{0}^{\infty }e^{-zx}L_{\alpha }(x)dx=e^{-z^{\alpha }},{\text{ where }}>\alpha <1.}
4061:
2842:
16325:
16315:
16006:
15932:
15633:
15492:
14837:
Weron, Rafał (1996). "On the Chambers-Mallows-Stuck method for simulating skewed stable random variables".
13964:
13896:
7260:
356:
16385:
10511:
The Lévy distribution of solar flare waiting time events (time between flare events) was demonstrated for
9253:
7256:
7244:
4742:
1288:
16370:
16365:
16310:
16246:
16190:
16011:
15998:
15789:
15734:
15686:
15477:
15406:
15271:
14735:
Chambers, J. M.; Mallows, C. L.; Stuck, B. W. (1976). "A Method for Simulating Stable Random Variables".
7236:
5824:
5395:
5306:
4688:
2418:
is the exponent or index of the distribution and specifies the asymptotic behavior of the distribution.
1317:
1294:
Of the four parameters defining the family, most attention has been focused on the stability parameter,
16504:
16280:
16099:
15881:
15834:
15703:
15679:
15659:
15502:
15376:
15256:
15028:
Garoni, T. M.; Frankel, N. E. (2002). "Lévy flights: Exact results and asymptotics beyond all orders".
5788:
1464:
1448:) random variables. The normal distribution defines a family of stable distributions. By the classical
615:
14050:
Mandelbrot, B. (1961). "Stable Paretian Random Functions and the Multiplicative Variation of Income".
10779:
132:
16509:
16293:
16254:
16128:
15965:
15809:
15754:
15652:
15616:
15487:
15452:
7430:
Log-log plot of symmetric centered stable distribution PDFs showing the power law behavior for large
6610:
5976:
5428:
4916:
997:
14988:
13944:
13523:
10237:
9941:
8868:
6556:
4850:
2910:
2621:
2022:
1679:, roughly corresponding to measures of asymmetry and concentration, respectively (see the figures).
1127:
16195:
15983:
15749:
15708:
15623:
15577:
15517:
15482:
15371:
15266:
15216:
14851:
14799:
Misiorek, Adam; Weron, Rafał (2012). Gentle, James E.; Härdle, Wolfgang Karl; Mori, Yuichi (eds.).
14525:
Gnedenko, Boris Vladimirovich; Kologorov, Andreĭ Nikolaevich; Doob, Joseph L.; Hsu, Pao-Lu (1968).
13940:
13934:
13920:
13005:
11488:
10721:
10588:
10452:
Stable distributions owe their importance in both theory and practice to the generalization of the
10436:
9469:
7259:. An English language version of the complete proof of the GCLT is available in the translation of
5919:
4032:
3699:
1628:
1417:
1283:
if its distribution is stable. The stable distribution family is also sometimes referred to as the
1250:
14808:. Springer Handbooks of Computational Statistics. Springer Berlin Heidelberg. pp. 1025–1059.
7514:
7507:
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large
7441:
6452:
3845:
3762:
3725:
3496:
3083:
2338:
2188:
1747:. The density function is therefore the inverse Fourier transform of the characteristic function:
1387:
387:
291:
230:
16494:
16436:
16107:
15894:
15804:
15759:
15744:
15562:
15512:
15507:
15308:
15288:
14800:
14702:
13879:
11727:
11693:
8041:
7704:
7240:
5952:
1688:
1502:
20:
15664:
2139:
16360:
16348:
16337:
16219:
16115:
15922:
15366:
15346:
15251:
14846:
14352:
Nolan, John P. (1997). "Numerical calculation of stable densities and distribution functions".
13916:
12996:
10386:
9710:
9293:
8099:
7978:
7925:
7637:
7572:
7476:
6620:
3791:
2433:
1353:
686:
577:
541:
505:
317:
11454:
10412:
8075:
8004:
7738:
7663:
6134:
5982:
5541:
5345:
3871:
2364:
2293:
465:
426:
256:
16484:
16441:
16285:
15960:
15814:
15794:
15691:
15261:
10472:
10453:
10088:
8579:. Reversing the order of integration and summation, and carrying out the integration yields:
7833:
7788:
7596:
6677:
6657:
5372:
3063:
2783:
2401:
2226:
1662:
1624:
all have the above property, it follows that they are special cases of stable distributions.
1449:
1297:
214:
124:
91:
45:
14587:
Zolotarev, V. (1995). "On Representation of Densities of Stable Laws by Special Functions".
14239:
10128:
7883:
7614:
6973:
6521:
2803:
2246:
1642:
171:
16534:
16529:
16524:
16519:
16456:
16426:
16305:
15948:
15839:
15442:
15401:
15396:
15293:
15156:
15075:
15037:
14918:
14634:
14409:
14312:. Springer Series in Operations Research and Financial Engineering. Switzerland: Springer.
10457:
8788:
8119:
7558:
7434:. The power law behavior is evidenced by the straight-line appearance of the PDF for large
6170:
6166:
2212:
15739:
14214:
11092:
11063:
10644:
10490:
10466:
10148:
9332:
8810:
For one-sided stable distribution, the above series expansion needs to be modified, since
7947:
7762:
5467:
1721:
1621:
8:
16593:
16468:
15973:
15943:
15917:
15871:
15799:
15611:
15547:
14462:"A Theory of Asset Return and Volatility Under Stable Law and Stable Lambda Distribution"
13864:
10715:
10582:
10505:
10440:
8028:
7957:
7932:
7915:
7687:
6614:
1617:
1613:
1377:
1347:
605:
15160:
15079:
15041:
14922:
14638:
14413:
16:
Distribution of variables which satisfies a stability property under linear combinations
16499:
15988:
15769:
15764:
15669:
15606:
15601:
15457:
15447:
15331:
15172:
14864:
14772:
14492:
14433:
14399:
14331:
14177:
14137:
14102:
14067:
14009:
13045:
10108:
9446:
9366:
7235:
The Generalized Central Limit Theorem (GCLT) was an effort of multiple mathematicians (
5535:
4665:
1272:
1258:
1246:
207:
1461:. In particular, he referred to those maximally skewed in the positive direction with
793:{\displaystyle \exp \!{\big (}t\mu -c^{\alpha }t^{\alpha }\sec(\pi \alpha /2){\big )}}
16397:
15824:
15567:
15497:
15462:
15411:
15091:
14994:
14969:
14934:
14889:
14860:
14817:
14779:
14752:
14650:
14604:
14547:
14425:
14369:
14335:
14321:
14281:
14032:
13869:
10494:
7810:
7806:
7264:
7252:
4891:
1715:
1454:
454:
15176:
14686:
14437:
14181:
6990:
variables it follows that these parameters for the convolved function are given by:
3484:{\displaystyle s=\left(\sum _{i=1}^{N}|k_{i}|^{\alpha }\right)^{\frac {1}{\alpha }}}
15572:
15246:
15185:
15164:
15083:
15045:
14961:
14926:
14868:
14856:
14809:
14748:
14744:
14717:
14681:
14669:
14642:
14625:
Peach, G. (1981). "Theory of the pressure broadening and shift of spectral lines".
14596:
14421:
14417:
14361:
14313:
14273:
14169:
14129:
14094:
14059:
14001:
13968:
13909:
11119:
5979:
is the first-order marginal distribution of a volatility process. In this context,
2398:> 0 is a scale factor which is a measure of the width of the distribution while
14813:
10819:
10461:
10085:
To simulate a stable random variable for all admissible values of the parameters
5069:
3166:{\displaystyle \delta -\beta \gamma \tan \left({\tfrac {\pi \alpha }{2}}\right).}
1527:
1458:
1276:
1265:
569:
198:
13856:
8144:
The stable distribution can be restated as the real part of a simpler integral:
15645:
15144:
14268:
Voit, Johannes (2005). Balian, R; Beiglböck, W; Grosse, H; Thirring, W (eds.).
9243:
7248:
3693:
2216:
376:
15168:
14965:
14721:
14646:
14543:
Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance
14365:
14317:
13908:
The STABLE program for Windows is available from John Nolan's stable webpage:
13851:
5072:
of the one-sided stable distribution. Its location-scale family is defined as
3837:
3759:" behavior causes the variance of stable distributions to be infinite for all
16577:
16268:
16016:
15303:
15087:
14973:
14930:
14756:
14654:
14608:
14373:
13992:
Mandelbrot, B. (1960). "The Pareto–Lévy Law and the Distribution of Income".
13874:
8329:
7557:). There are, however three special cases which can be expressed in terms of
6606:
1997:
14307:
7503:
3110:, whereas in the second parametrization when the mean exists it is equal to
2287:
as it should be so that the probability distribution function will be real.
15095:
14429:
14036:
10501:
8783:
and will converge for appropriate values of the parameters. (Note that the
2892:. In the first parametrization, this is done by defining the new variable:
14938:
6449:", or the "generalized error/normal distribution", often referred to when
3814:, the distribution is Gaussian (see below), with tails asymptotic to exp(−
14571:
14120:
Fama, Eugene F. (1963). "Mandelbrot and the Stable Paretian Hypothesis".
13887:
Financial models with long-tailed distributions and volatility clustering
8118:
approaches zero or as α approaches zero the distribution will approach a
7785:
A general closed form expression for stable PDFs with rational values of
6647:
1457:
referred to such distributions as "stable Paretian distributions", after
7817:
and those that are expressible by special functions are indicated by an
7426:
14496:
14480:
14141:
14106:
14071:
14013:
10469:
to propose that cotton prices follow an alpha-stable distribution with
6654:
Stable distributions are closed under convolution for a fixed value of
4840:{\textstyle {\frac {1}{\nu }}L_{\alpha }\left({\frac {x}{\nu }}\right)}
3756:
1444:" for properly normed sums of independent and identically distributed (
1440:. The importance of stable probability distributions is that they are "
15049:
14085:
Mandelbrot, B. (1963). "The Variation of Certain Speculative Prices".
8319:{\displaystyle f(x;\alpha ,\beta ,c,\mu )={\frac {1}{\pi }}\Re \left.}
1828:{\displaystyle \varphi (t)=\int _{-\infty }^{\infty }f(x)e^{ixt}\,dx.}
8768:{\displaystyle f(x;\alpha ,\beta ,c,\mu )={\frac {1}{\pi }}\Re \left}
8518:{\displaystyle f(x;\alpha ,\beta ,c,\mu )={\frac {1}{\pi }}\Re \left}
5527:{\displaystyle {\mathfrak {N}}_{\frac {1}{2}}(\nu ;\nu _{0},\theta )}
4186:{\displaystyle \varphi (t;\alpha )=\exp \left(-q|t|^{\alpha }\right)}
1441:
14885:
Simulation and Chaotic Behavior of Alpha-stable Stochastic Processes
14600:
14461:
14277:
14063:
14005:
14173:
14133:
14098:
13930:
9433:{\displaystyle \left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)}
2182:
2181:, is a measure of asymmetry. Notice that in this context the usual
1381:
533:
493:
189:
14404:
5909:{\textstyle {\mathfrak {N}}_{\frac {1}{2}}(\nu ;\nu _{0},\theta )}
1841:
is called stable if its characteristic function can be written as
1505:
is a stable distribution if it satisfies the following property:
610:
not analytically expressible, except for certain parameter values
371:
not analytically expressible, except for certain parameter values
14703:"Simple consistent estimators of stable distribution parameters"
6165:, and one can decompose the integral on the left hand side as a
14158:
Mandelbrot, B. (1963). "New methods in statistical economics".
2456:. A continuous parametrization, better for numerical work, is
415:
10456:
to random variables without second (and possibly first) order
1627:
Such distributions form a four-parameter family of continuous
922:{\displaystyle \exp \!{\big (}t\mu -c2\pi ^{-1}t\ln t{\big )}}
361:
not analytically expressible, except for some parameter values
14309:
Univariate stable distributions, Models for Heavy Tailed Data
13958:
3080:. In the first parametrization, if the mean exists (that is,
1269:
14527:
Limit distributions for sums of independent random variables
13824:
10512:
10321:
10011:
3838:
One-sided stable distribution and stable count distribution
3785:. This property is illustrated in the log–log plots below.
3751:, the tail does not vanish to the left or right, resp., of
3003:
2764:
2115:
1411:
1226:
72:
Skewed centered stable distributions with unit scale factor
15014:
Leddon, D., A statistical Study of Hard X-Ray Solar Flares
14573:
Continuous and discrete properties of stochastic processes
4662:
The double-sine integral is more effective for very small
15143:
Zaliapin, I. V.; Kagan, Y. Y.; Schoenberg, F. P. (2005).
14270:
The Statistical Mechanics of Financial Markets – Springer
7971:
Some of the special cases are known by particular names:
5849:
1445:
14710:
Communications in Statistics. Simulation and Computation
14670:"Representation of e^{-x^\lambda} As a Laplace Integral"
14524:
8035:) which has a specific usage in physics under this name.
7230:
5293:{\displaystyle \theta >0,{\text{ and }}\alpha <1.}
3391:{\displaystyle {\tfrac {1}{s}}f(y/s;\alpha ,\beta ,c,0)}
15142:
676:{\displaystyle \exp \!{\big (}t\mu +c^{2}t^{2}{\big )}}
13477:
13429:
13364:
13277:
13229:
13161:
13074:
12939:
12921:
12906:
12891:
12876:
12861:
12796:
12727:
12709:
12694:
12679:
12664:
12649:
12628:
12536:
12518:
12503:
12488:
12473:
12458:
12412:
12360:
12299:
12284:
12269:
12254:
12239:
12196:
12173:
12148:
12125:
12016:
12001:
11986:
11971:
11956:
11913:
11890:
11865:
11842:
11756:
11513:
11396:
11340:
11288:
11232:
11144:
10844:
10667:
9414:
9399:
7549:
There is no general analytic solution for the form of
5858:
5431:
5398:
4894:
4853:
4798:
4745:
4691:
3333:
3140:
2680:
1177:
1137:
15065:
14208:"Stable Distributions – Models for Heavy Tailed Data"
13054:
13008:
11736:
11696:
11497:
11457:
11128:
11095:
11066:
10828:
10782:
10724:
10653:
10591:
10530:
10475:
10415:
10389:
10341:
10225:
10171:
10151:
10131:
10111:
10091:
10034:
9892:
9739:
9713:
9498:
9472:
9449:
9389:
9369:
9335:
9296:
9256:
8907:
8871:
8816:
8585:
8531:
8338:
8150:
8078:
8044:
8007:
7981:
7886:
7836:
7791:
7741:
7707:
7666:
7640:
7617:
7575:
7517:
7511:. Again the slope of the linear portions is equal to
7479:
7444:
6996:
6976:
6700:
6680:
6660:
6623:
6559:
6524:
6484:
6455:
6182:
6137:
6018:
5985:
5955:
5922:
5827:
5791:
5570:
5544:
5476:
5375:
5348:
5309:
5263:
5081:
4929:
4668:
4237:
4199:
4119:
4064:
4035:
3915:
3874:
3848:
3794:
3765:
3728:
3702:
3525:
3499:
3404:
3331:
3251:
3186:
3116:
3086:
3066:
3025:
2898:
2845:
2806:
2786:
2609:
2462:
2436:
2404:
2367:
2341:
2296:
2249:
2229:
2191:
2142:
2010:
1847:
1753:
1724:
1691:
1665:
1645:
1467:
1420:
1390:
1356:
1320:
1300:
1115:
1000:
935:
860:
806:
716:
689:
625:
580:
544:
508:
468:
429:
390:
320:
294:
259:
233:
174:
135:
94:
48:
15215:
14734:
14385:
14383:
16589:Probability distributions with non-finite variance
14771:
14539:
13834:
13036:
12987:
11718:
11680:
11479:
11441:
11110:
11081:
11050:
10810:
10766:
10704:
10633:
10569:
10481:
10427:
10401:
10375:
10327:
10211:
10157:
10137:
10117:
10097:
10074:
10017:
9878:
9725:
9697:
9484:
9455:
9432:
9375:
9350:
9321:
9290:There are no analytic expressions for the inverse
9274:
9227:
8893:
8857:
8767:
8571:
8517:
8318:
8090:
8064:
8019:
7993:
7892:
7842:
7797:
7753:
7727:
7678:
7652:
7623:
7587:
7538:
7491:
7465:
7217:
6982:
6948:
6686:
6666:
6635:
6581:
6545:
6510:
6467:
6433:
6157:
6120:
5998:
5967:
5941:
5908:
5840:
5813:
5774:
5553:
5526:
5458:
5417:
5381:
5361:
5334:
5292:
5249:
5057:
4906:
4880:
4839:
4780:
4731:
4674:
4652:
4223:
4185:
4105:
4047:
4021:
3886:
3860:
3806:
3777:
3743:
3714:
3684:
3511:
3483:
3390:
3314:{\displaystyle Y=\sum _{i=1}^{N}k_{i}(X_{i}-\mu )}
3313:
3225:
3165:
3098:
3072:
3052:
3009:
2884:
2812:
2792:
2770:
2595:
2448:
2410:
2379:
2353:
2308:
2255:
2235:
2203:
2169:
2121:
1980:
1827:
1739:
1706:
1671:
1651:
1485:
1432:
1402:
1368:
1338:
1306:
1268:with this distribution has the same distribution,
1232:
1099:
974:
921:
845:
792:
701:
675:
592:
556:
520:
480:
441:
402:
335:
306:
271:
245:
180:
159:
100:
54:
15108:
5303:It is also a one-sided distribution supported on
1089:
1007:
1004:
864:
720:
629:
16575:
14380:
13910:http://www.robustanalysis.com/public/stable.html
9285:
15145:"Approximating the Distribution of Pareto Sums"
15061:
15059:
14951:
14882:Janicki, Aleksander; Weron, Aleksander (1994).
14737:Journal of the American Statistical Association
14512:Theorie de l'addition des variables aleatoires
14354:Communications in Statistics. Stochastic Models
6511:{\displaystyle {\mathfrak {N}}_{\alpha }(\nu )}
2215:, and the usual skewness definition is the 3rd
14987:Rachev, Svetlozar T.; Mittnik, Stefan (2000).
14802:Heavy-Tailed Distributions in VaR Calculations
10504:as a general expression for a quasistatically
4193:. Thus the integral form of its PDF is (note:
3053:{\displaystyle y={\frac {x-\delta }{\gamma }}}
2211:the distribution does not admit 2nd or higher
1631:parametrized by location and scale parameters
1291:, the first mathematician to have studied it.
846:{\displaystyle \alpha \neq 1,\beta =-1,t>0}
15201:
15102:
15027:
15023:
15021:
14986:
14798:
14347:
14345:
14272:. Texts and Monographs in Physics. Springer.
10212:{\displaystyle X\sim S_{\alpha }(\beta ,1,0)}
10075:{\displaystyle X\sim S_{\alpha }(\beta ,1,0)}
8572:{\displaystyle q=c^{\alpha }(1-i\beta \Phi )}
5470:which is an inverse gamma distribution. Thus
3755:, although the above expression is 0). This "
914:
867:
785:
723:
668:
632:
118:CDFs for skewed centered stable distributions
15056:
14881:
14389:
14301:
14299:
14297:
7499:, in black, which is a normal distribution.)
1346:, with the upper bound corresponding to the
1279:parameters. A random variable is said to be
62:-stable distributions with unit scale factor
13902:
10705:{\displaystyle f(x;{\tfrac {1}{2}},1,1,0).}
10435:) the CMS method reduces to the well known
4224:{\displaystyle \operatorname {Im} (q)<0}
3019:For the second parametrization, simply use
15208:
15194:
15018:
14589:Theory of Probability and Its Applications
14342:
14157:
14084:
14049:
13991:
10570:{\displaystyle f(x;\alpha ,\beta ,c,\mu )}
10376:{\displaystyle S_{\alpha }(\beta ,c,\mu )}
6173:and a standard stable count distribution,
4920:. Its standard distribution is defined as
3493:The asymptotic behavior is described, for
3226:{\displaystyle f(x;\alpha ,\beta ,c,\mu )}
2283:is the complex conjugate of its value at −
1086:
1012:
975:{\displaystyle \alpha =1,\beta =-1,t>0}
14850:
14769:
14700:
14685:
14586:
14403:
14294:
14153:
14151:
13987:
13985:
9359:formula yielded the following algorithm:
8503:
8301:
7270:The statement of the GLCT is as follows:
6336:
4636:
4622:
4567:
4495:
4481:
4423:
4353:
1815:
1639:, respectively, and two shape parameters
1049:
1033:
15123:
14908:
10028:This algorithm yields a random variable
8139:
7502:
7425:
2316:, the characteristic function is just a
488:, otherwise not analytically expressible
449:, otherwise not analytically expressible
14667:
14576:. PhD thesis, University of Nottingham.
14540:Samorodnitsky, G.; Taqqu, M.S. (1994).
14481:"The Central Limit Theorem around 1935"
10519:
9237:
8858:{\displaystyle q=\exp(-i\alpha \pi /2)}
8328:Expressing the second exponential as a
6589:, and all positive moments are finite.
4106:{\displaystyle q=\exp(-i\alpha \pi /2)}
2885:{\displaystyle f(y;\alpha ,\beta ,1,0)}
2320:; the distribution is symmetric about
1714:of any probability distribution is the
1314:(see panel). Stable distributions have
16576:
14478:
14148:
13982:
6646:Stable distributions are closed under
15189:
14836:
14624:
14620:
14618:
14565:
14563:
14455:
14453:
14451:
14449:
14447:
14351:
14305:
7231:The Generalized Central Limit Theorem
5068:The stable count distribution is the
4781:{\textstyle X_{i}\sim L_{\alpha }(x)}
16558:
14839:Statistics & Probability Letters
14774:One-Dimensional Stable Distributions
14509:
14459:
14267:
14263:
14261:
14259:
14237:
14201:
14199:
14197:
14195:
14193:
14191:
14119:
14026:
10493:are frequently found in analysis of
9275:{\displaystyle 0.5<\alpha \leq 2}
7561:as can be seen by inspection of the
3894:, the distribution is supported on [
3241:is the sum of independent copies of
2387:, the distribution is supported on [
1718:of its probability density function
14569:
8787: = 0 term which yields a
6488:
5862:
5841:{\displaystyle {\sqrt {24}}\theta }
5574:
5480:
5418:{\textstyle \alpha ={\frac {1}{2}}}
5085:
4933:
4732:{\textstyle Y=\sum _{i=1}^{N}X_{i}}
3175:
2421:
2326:symmetric alpha-stable distribution
2263:, but possibly different values of
1380:. The distributions have undefined
188:∈ — skewness parameter (note that
13:
14615:
14560:
14444:
12788:
12404:
12188:
12165:
12140:
12117:
11905:
11882:
11857:
11834:
10881:
9200:
9111:
9049:
8972:
8948:
8739:
8659:
8635:
8563:
8455:
8406:
8388:
8293:
8218:
8200:
6938:
6874:
6362:
6271:
6193:
6029:
6006:is called the "floor volatility".
5335:{\displaystyle [\nu _{0},\infty )}
5326:
5133:
4962:
4540:
4395:
4299:
4294:
4278:
4113:, its characteristic function is
3653:
2610:
2580:
2011:
1965:
1782:
1777:
1339:{\displaystyle 0<\alpha \leq 2}
1116:
1080:
14:
16615:
14990:Stable Paretian Models in Finance
14778:. American Mathematical Society.
14256:
14205:
14188:
5814:{\displaystyle \nu _{0}+6\theta }
1598:. The distribution is said to be
1526:be independent realizations of a
1486:{\displaystyle 1<\alpha <2}
16557:
16548:
16547:
14161:The Journal of Political Economy
13892:Multivariate stable distribution
13861:Other "power law" distributions
10811:{\displaystyle S_{\mu ,\nu }(z)}
10506:pressure broadened spectral line
8807:which is generally less useful.
7421:
7275:A non-degenerate random variable
5459:{\textstyle L_{\frac {1}{2}}(x)}
2276:cumulative distribution function
160:{\displaystyle \alpha \in (0,2]}
111:
81:
79:Cumulative distribution function
65:
35:
15136:
15117:
15030:Journal of Mathematical Physics
15007:
14980:
14945:
14902:
14875:
14830:
14792:
14763:
14728:
14694:
14687:10.1090/S0002-9904-1946-08672-3
14661:
14580:
14533:
14518:
14503:
14472:
10447:
10165:use the following property: If
9929:
7418:must be a stable distribution.
5762:
5369:is the cut-off location, while
4881:{\textstyle \nu =N^{1/\alpha }}
2324:and is referred to as a (Lévy)
1100:{\displaystyle \exp \!{\Big },}
14749:10.1080/01621459.1976.10480344
14529:. Reading, MA: Addison-wesley.
14422:10.1103/PhysRevLett.105.210604
14231:
14113:
14078:
14043:
14020:
13704:
13696:
13550:
13542:
13342:
13334:
13142:
13134:
13031:
13025:
11474:
11468:
11417:
11409:
11358:
11350:
11309:
11301:
11250:
11242:
11203:
11194:
11105:
11099:
11076:
11070:
10805:
10799:
10758:
10728:
10696:
10657:
10625:
10595:
10564:
10534:
10370:
10352:
10206:
10188:
10069:
10051:
9969:
9960:
9660:
9657:
9645:
9633:
9602:
9598:
9592:
9583:
9578:
9575:
9563:
9557:
9345:
9339:
9316:
9310:
9218:
9203:
9152:
9146:
9134:
9128:
9067:
9052:
8990:
8980:
8928:
8922:
8894:{\displaystyle qi^{\alpha }=1}
8852:
8829:
8757:
8742:
8677:
8667:
8619:
8589:
8566:
8548:
8483:
8463:
8434:
8422:
8372:
8342:
8296:
8278:
8269:
8259:
8246:
8234:
8184:
8154:
8098:the distribution reduces to a
7761:the distribution reduces to a
7686:the distribution reduces to a
7595:the distribution reduces to a
7533:
7521:
7460:
7448:
6935:
6929:
6913:
6894:
6871:
6865:
6849:
6830:
6803:
6784:
6770:
6751:
6582:{\displaystyle L_{\alpha }(x)}
6576:
6570:
6540:
6528:
6505:
6499:
6447:exponential power distribution
6402:
6393:
6378:
6365:
6287:
6274:
6243:
6235:
6066:
6060:
5975:(See Section 7 of ). Thus the
5903:
5878:
5821:and its standard deviation is
5678:
5658:
5615:
5590:
5521:
5496:
5453:
5447:
5329:
5310:
5149:
5136:
5121:
5096:
4950:
4944:
4775:
4769:
4633:
4619:
4613:
4604:
4595:
4586:
4564:
4558:
4492:
4478:
4472:
4460:
4451:
4442:
4420:
4414:
4341:
4332:
4258:
4252:
4212:
4206:
4168:
4159:
4135:
4123:
4100:
4077:
3932:
3926:
3906: = 0) is defined as
3668:
3656:
3618:
3612:
3606:
3591:
3557:
3548:
3535:
3529:
3455:
3439:
3385:
3347:
3308:
3289:
3220:
3190:
2879:
2849:
2746:
2735:
2642:
2630:
2577:
2571:
2538:
2526:
2496:
2466:
2318:stretched exponential function
2164:
2149:
2097:
2089:
1962:
1956:
1923:
1911:
1881:
1851:
1796:
1790:
1763:
1757:
1734:
1728:
1701:
1695:
1285:Lévy alpha-stable distribution
1203:
1195:
1083:
1077:
1071:
1050:
1039:
1026:
780:
763:
154:
142:
1:
13994:International Economic Review
13976:
13953:implementation is located in
13037:{\displaystyle W_{k,\mu }(z)}
10767:{\displaystyle f(x;2,0,1,0).}
10634:{\displaystyle f(x;1,0,1,0).}
9485:{\displaystyle \alpha \neq 1}
9286:Simulation of stable variates
7473:. (The only exception is for
6592:
5942:{\displaystyle \nu _{0}=10.4}
4048:{\displaystyle \alpha <1.}
3902:. Its standard distribution (
3900:one-sided stable distribution
3715:{\displaystyle \alpha \geq 1}
1575:has the same distribution as
1496:
1433:{\displaystyle \alpha \leq 1}
14861:10.1016/0167-7152(95)00113-1
14814:10.1007/978-3-642-21551-3_34
14701:McCulloch, J Huston (1986).
14479:Le Cam, L. (February 1986).
13897:Discrete-stable distribution
13880:Zipf–Mandelbrot distribution
10082:. For a detailed proof see.
9242:In addition to the existing
7539:{\displaystyle -(\alpha +1)}
7466:{\displaystyle -(\alpha +1)}
6613:, with the exception of the
6468:{\displaystyle \alpha >1}
3898:, ∞). This family is called
3861:{\displaystyle \alpha <1}
3778:{\displaystyle \alpha <2}
3744:{\displaystyle \beta =\pm 1}
3512:{\displaystyle \alpha <2}
3099:{\displaystyle \alpha >1}
2354:{\displaystyle \alpha <1}
2204:{\displaystyle \alpha <2}
2185:is not well defined, as for
1403:{\displaystyle \alpha <2}
403:{\displaystyle \alpha >1}
307:{\displaystyle \alpha <1}
246:{\displaystyle \alpha <1}
33:Probability density function
7:
15149:Pure and Applied Geophysics
14031:. Paris: Gauthier-Villars.
13845:
11719:{\displaystyle {}_{m}F_{n}}
9442:exponential random variable
9363:generate a random variable
8065:{\displaystyle \alpha =3/2}
7728:{\displaystyle \alpha =1/2}
7402:at all continuity points of
5968:{\displaystyle \theta =1.6}
1707:{\displaystyle \varphi (t)}
10:
16620:
16381:Wrapped asymmetric Laplace
15352:Extended negative binomial
14514:. Paris: Gauthier-Villars.
12995:with the latter being the
11491:of the second kind, then:
7438:, with the slope equal to
6611:heavy-tailed distributions
5848:. It is hypothesized that
2828:) should be positive, and
2170:{\displaystyle \beta \in }
18:
16543:
16477:
16435:
16336:
16172:
16150:
16141:
16040:Generalized extreme value
16025:
15860:
15820:Relativistic Breit–Wigner
15536:
15433:
15424:
15317:
15237:
15228:
15217:Probability distributions
15169:10.1007/s00024-004-2666-3
14966:10.1080/02331889208802383
14770:Zolotarev, V. M. (1986).
14722:10.1080/03610918608812563
14647:10.1080/00018738100101467
14366:10.1080/15326349708807450
14318:10.1007/978-3-030-52915-4
10402:{\displaystyle \alpha =2}
9726:{\displaystyle \alpha =1}
9383:uniformly distributed on
9322:{\displaystyle F^{-1}(x)}
7994:{\displaystyle \alpha =1}
7930:
7880:
7830:
7805:is available in terms of
7653:{\displaystyle \alpha =1}
7611:; the skewness parameter
7588:{\displaystyle \alpha =2}
7492:{\displaystyle \alpha =2}
6962:is not a function of the
6636:{\displaystyle \alpha =2}
6605:Stable distributions are
6598:Stable distributions are
5977:stable count distribution
5342:. The location parameter
4917:stable count distribution
3807:{\displaystyle \alpha =2}
2449:{\displaystyle \alpha =1}
1629:probability distributions
1369:{\displaystyle \alpha =1}
993:
988:
702:{\displaystyle \alpha =2}
619:
614:
609:
604:
593:{\displaystyle \alpha =2}
573:
568:
557:{\displaystyle \alpha =2}
537:
532:
521:{\displaystyle \alpha =2}
497:
492:
458:
453:
419:
414:
380:
375:
370:
365:
360:
355:
336:{\displaystyle \beta =-1}
218:
213:
128:
123:
77:
31:
16584:Continuous distributions
15088:10.1103/physreve.60.5327
14931:10.1103/PhysRevE.49.4677
14668:Pollard, Howard (1946).
13903:Software implementations
11728:hypergeometric functions
11489:modified Bessel function
11480:{\displaystyle K_{v}(x)}
10428:{\displaystyle \beta =0}
8775:which will be valid for
8091:{\displaystyle \beta =0}
8027:, the distribution is a
8020:{\displaystyle \beta =1}
7754:{\displaystyle \beta =1}
7679:{\displaystyle \beta =0}
6158:{\displaystyle x=1/\nu }
5999:{\displaystyle \nu _{0}}
5554:{\displaystyle 4\theta }
5362:{\displaystyle \nu _{0}}
3887:{\displaystyle \beta =1}
2820:are the same as before,
2380:{\displaystyle \beta =1}
2309:{\displaystyle \beta =0}
481:{\displaystyle \beta =0}
442:{\displaystyle \beta =0}
272:{\displaystyle \beta =1}
19:Not to be confused with
16604:Stability (probability)
16035:Generalized chi-squared
15979:Normal-inverse Gaussian
14392:Physical Review Letters
14306:Nolan, John P. (2020).
14122:The Journal of Business
14087:The Journal of Business
14029:Calcul des probabilités
13955:scipy.stats.levy_stable
10500:They are also found in
10482:{\displaystyle \alpha }
10098:{\displaystyle \alpha }
7843:{\displaystyle \alpha }
7798:{\displaystyle \alpha }
7563:characteristic function
6687:{\displaystyle \alpha }
6667:{\displaystyle \alpha }
5538:of shape 3/2 and scale
5382:{\displaystyle \theta }
3233:denotes the density of
3073:{\displaystyle \alpha }
2793:{\displaystyle \alpha }
2411:{\displaystyle \alpha }
2236:{\displaystyle \alpha }
1684:characteristic function
1672:{\displaystyle \alpha }
1503:degenerate distribution
1307:{\displaystyle \alpha }
101:{\displaystyle \alpha }
55:{\displaystyle \alpha }
21:Stationary distribution
16347:Univariate (circular)
15908:Generalized hyperbolic
15337:Conway–Maxwell–Poisson
15327:Beta negative binomial
14460:Lihn, Stephen (2017).
14244:www.randomservices.org
14240:"Stable Distributions"
13969:StableDistributions.jl
13917:GNU Scientific Library
13836:
13038:
12997:Holtsmark distribution
12989:
11720:
11682:
11481:
11443:
11112:
11083:
11052:
10812:
10768:
10706:
10635:
10571:
10483:
10429:
10403:
10377:
10329:
10213:
10159:
10139:
10138:{\displaystyle \beta }
10119:
10099:
10076:
10019:
9880:
9727:
9699:
9486:
9457:
9434:
9377:
9352:
9323:
9276:
9229:
9115:
8976:
8895:
8859:
8769:
8663:
8573:
8519:
8459:
8320:
8114:Also, in the limit as
8100:Holtsmark distribution
8092:
8066:
8021:
7995:
7894:
7893:{\displaystyle \beta }
7844:
7799:
7755:
7729:
7680:
7654:
7625:
7624:{\displaystyle \beta }
7589:
7546:
7540:
7500:
7493:
7467:
7219:
6984:
6983:{\displaystyle \beta }
6950:
6688:
6668:
6637:
6583:
6547:
6546:{\displaystyle -(n+1)}
6512:
6469:
6435:
6159:
6122:
6000:
5969:
5943:
5910:
5842:
5815:
5776:
5555:
5528:
5460:
5419:
5383:
5363:
5336:
5294:
5251:
5059:
4908:
4882:
4841:
4782:
4733:
4718:
4685:Consider the Lévy sum
4676:
4654:
4225:
4187:
4107:
4049:
4023:
3888:
3862:
3808:
3779:
3745:
3716:
3686:
3513:
3485:
3437:
3392:
3315:
3278:
3227:
3167:
3106:) then it is equal to
3100:
3074:
3054:
3011:
2886:
2814:
2813:{\displaystyle \beta }
2794:
2772:
2597:
2450:
2412:
2381:
2355:
2310:
2257:
2256:{\displaystyle \beta }
2237:
2205:
2171:
2136:is a shift parameter,
2123:
1982:
1829:
1741:
1708:
1673:
1653:
1652:{\displaystyle \beta }
1487:
1434:
1404:
1370:
1340:
1308:
1234:
1101:
976:
923:
847:
794:
703:
677:
594:
558:
522:
482:
443:
404:
337:
308:
273:
247:
182:
181:{\displaystyle \beta }
167:— stability parameter
161:
102:
56:
16392:Bivariate (spherical)
15890:Kaniadakis κ-Gaussian
14674:Bull. Amer. Math. Soc
13837:
13039:
12990:
11721:
11683:
11482:
11444:
11113:
11084:
11053:
10813:
10769:
10707:
10636:
10572:
10484:
10460:and the accompanying
10454:central limit theorem
10430:
10404:
10378:
10330:
10214:
10160:
10140:
10120:
10100:
10077:
10020:
9881:
9728:
9700:
9487:
9458:
9435:
9378:
9353:
9324:
9277:
9230:
9095:
8956:
8896:
8860:
8770:
8643:
8574:
8520:
8439:
8321:
8140:Series representation
8102:with scale parameter
8093:
8067:
8022:
7996:
7895:
7845:
7800:
7765:with scale parameter
7756:
7730:
7690:with scale parameter
7681:
7655:
7626:
7597:Gaussian distribution
7590:
7541:
7506:
7494:
7468:
7429:
7220:
6985:
6951:
6689:
6669:
6638:
6584:
6548:
6513:
6470:
6436:
6160:
6123:
6001:
5970:
5944:
5911:
5843:
5816:
5777:
5556:
5529:
5461:
5420:
5384:
5364:
5337:
5295:
5252:
5060:
4909:
4883:
4842:
4783:
4734:
4698:
4677:
4655:
4226:
4188:
4108:
4050:
4024:
3889:
3863:
3809:
3780:
3746:
3717:
3687:
3514:
3486:
3417:
3393:
3316:
3258:
3228:
3168:
3101:
3075:
3055:
3012:
2887:
2815:
2795:
2773:
2598:
2451:
2413:
2382:
2356:
2311:
2290:In the simplest case
2258:
2238:
2206:
2172:
2124:
1983:
1830:
1742:
1709:
1674:
1654:
1545:if for any constants
1488:
1450:central limit theorem
1435:
1405:
1371:
1341:
1309:
1235:
1102:
977:
924:
848:
795:
704:
678:
600:, otherwise undefined
595:
564:, otherwise undefined
559:
523:
483:
444:
410:, otherwise undefined
405:
338:
309:
274:
248:
183:
162:
108:-stable distributions
103:
57:
16599:Stable distributions
16457:Dirac delta function
16404:Bivariate (toroidal)
16361:Univariate von Mises
16232:Multivariate Laplace
16124:Shifted log-logistic
15473:Continuous Bernoulli
14570:Lee, Wai Ha (2010).
13919:which is written in
13052:
13006:
11734:
11694:
11495:
11455:
11126:
11111:{\displaystyle C(x)}
11093:
11082:{\displaystyle S(x)}
11064:
10826:
10780:
10722:
10651:
10589:
10528:
10520:Other analytic cases
10497:and financial data.
10473:
10437:Box-Muller transform
10413:
10387:
10339:
10223:
10169:
10158:{\displaystyle \mu }
10149:
10129:
10109:
10089:
10032:
9890:
9737:
9711:
9496:
9470:
9447:
9387:
9367:
9351:{\displaystyle F(x)}
9333:
9294:
9254:
9248:parameter estimation
9238:Parameter estimation
8905:
8869:
8814:
8583:
8529:
8336:
8148:
8120:Dirac delta function
8106:and shift parameter
8076:
8042:
8005:
7979:
7884:
7834:
7789:
7769:and shift parameter
7739:
7705:
7694:and shift parameter
7664:
7638:
7615:
7573:
7559:elementary functions
7515:
7477:
7442:
6994:
6974:
6698:
6678:
6658:
6621:
6600:infinitely divisible
6557:
6522:
6482:
6453:
6180:
6171:Laplace distribution
6167:product distribution
6135:
6016:
5983:
5953:
5920:
5856:
5852:is distributed like
5825:
5789:
5568:
5542:
5474:
5429:
5396:
5373:
5346:
5307:
5261:
5079:
4927:
4892:
4851:
4796:
4743:
4689:
4666:
4235:
4197:
4117:
4062:
4033:
3913:
3872:
3846:
3792:
3763:
3726:
3700:
3523:
3497:
3402:
3329:
3249:
3184:
3114:
3084:
3064:
3023:
2896:
2843:
2804:
2784:
2607:
2460:
2434:
2402:
2365:
2339:
2328:, often abbreviated
2294:
2247:
2227:
2189:
2140:
2008:
1845:
1751:
1740:{\displaystyle f(x)}
1722:
1689:
1663:
1643:
1559:the random variable
1465:
1418:
1388:
1354:
1318:
1298:
1113:
998:
933:
858:
804:
714:
687:
623:
578:
542:
528:, otherwise infinite
506:
466:
427:
388:
318:
292:
257:
231:
172:
133:
92:
46:
16505:Natural exponential
16410:Bivariate von Mises
16376:Wrapped exponential
16242:Multivariate stable
16237:Multivariate normal
15558:Benktander 2nd kind
15553:Benktander 1st kind
15342:Discrete phase-type
15161:2005PApGe.162.1187Z
15080:1999PhRvE..60.5327H
15042:2002JMP....43.2670G
14923:1994PhRvE..49.4677M
14639:1981AdPhy..30..367P
14627:Advances in Physics
14510:Lévy, Paul (1937).
14485:Statistical Science
14414:2010PhRvL.105u0604P
14027:Lévy, Paul (1925).
13865:Pareto distribution
10716:Normal distribution
10583:Cauchy Distribution
9440:and an independent
9244:tests for normality
8410:
8222:
8029:Landau distribution
7688:Cauchy distribution
7207:
7189:
7172:
7144:
7086:
7068:
6615:normal distribution
6478:The n-th moment of
6197:
6033:
5389:defines its scale.
4544:
4399:
4303:
1786:
1618:Cauchy distribution
1614:normal distribution
1602:if this holds with
1585:for some constants
1378:Cauchy distribution
1348:normal distribution
984:otherwise undefined
88:CDFs for symmetric
28:
16160:Rectified Gaussian
16045:Generalized Pareto
15903:Generalized normal
15775:Matrix-exponential
13832:
13830:
13823:
13486:
13438:
13373:
13286:
13238:
13170:
13083:
13046:Whittaker function
13034:
12985:
12983:
12974:
12930:
12915:
12900:
12885:
12870:
12805:
12762:
12718:
12703:
12688:
12673:
12658:
12637:
12571:
12527:
12512:
12497:
12482:
12467:
12421:
12369:
12334:
12293:
12278:
12263:
12248:
12205:
12182:
12157:
12134:
12051:
12010:
11995:
11980:
11965:
11922:
11899:
11874:
11851:
11765:
11716:
11678:
11522:
11477:
11439:
11423:
11364:
11315:
11256:
11153:
11108:
11079:
11048:
10853:
10808:
10764:
10702:
10676:
10631:
10567:
10491:Lévy distributions
10479:
10425:
10399:
10373:
10325:
10320:
10209:
10155:
10135:
10115:
10095:
10072:
10015:
10010:
9876:
9723:
9695:
9482:
9453:
9430:
9423:
9408:
9373:
9348:
9319:
9272:
9225:
9223:
8891:
8855:
8765:
8569:
8515:
8396:
8316:
8208:
8088:
8062:
8017:
7991:
7890:
7840:
7807:Meijer G-functions
7795:
7751:
7725:
7676:
7650:
7621:
7585:
7547:
7536:
7501:
7489:
7463:
7215:
7213:
7193:
7175:
7158:
7130:
7072:
7054:
6980:
6946:
6684:
6664:
6633:
6579:
6543:
6508:
6465:
6431:
6183:
6155:
6118:
6019:
5996:
5965:
5939:
5906:
5838:
5811:
5772:
5551:
5536:gamma distribution
5524:
5456:
5415:
5379:
5359:
5332:
5290:
5247:
5055:
4904:
4878:
4837:
4778:
4729:
4672:
4650:
4648:
4530:
4385:
4286:
4221:
4183:
4103:
4045:
4019:
3884:
3858:
3804:
3775:
3741:
3712:
3696:(except that when
3682:
3509:
3481:
3388:
3342:
3311:
3223:
3163:
3154:
3096:
3070:
3050:
3007:
3002:
2882:
2836:) should be real.
2810:
2790:
2768:
2763:
2694:
2593:
2446:
2408:
2377:
2351:
2306:
2253:
2233:
2201:
2179:skewness parameter
2167:
2119:
2114:
1978:
1825:
1769:
1737:
1704:
1669:
1649:
1483:
1430:
1400:
1366:
1336:
1304:
1259:linear combination
1247:probability theory
1230:
1225:
1186:
1151:
1097:
972:
919:
843:
790:
699:
673:
590:
554:
518:
478:
439:
400:
333:
304:
269:
243:
208:location parameter
178:
157:
98:
52:
26:
16571:
16570:
16168:
16167:
16137:
16136:
16028:whose type varies
15974:Normal (Gaussian)
15928:Hyperbolic secant
15877:Exponential power
15780:Maxwell–Boltzmann
15528:Wigner semicircle
15420:
15419:
15392:Parabolic fractal
15382:Negative binomial
15068:Physical Review E
15050:10.1063/1.1467095
15000:978-0-471-95314-2
14911:Physical Review E
14823:978-3-642-21550-6
14785:978-0-8218-4519-6
14327:978-3-030-52914-7
14287:978-3-540-26285-5
13967:provides package
13875:Zipf distribution
13870:Zeta distribution
13793:
13776:
13763:
13730:
13709:
13693:
13683:
13639:
13619:
13606:
13576:
13555:
13539:
13532:
13485:
13437:
13419:
13406:
13372:
13347:
13331:
13324:
13285:
13237:
13219:
13206:
13169:
13147:
13131:
13121:
13082:
12973:
12929:
12914:
12899:
12884:
12869:
12834:
12804:
12761:
12717:
12702:
12687:
12672:
12657:
12636:
12601:
12570:
12526:
12511:
12496:
12481:
12466:
12431:
12420:
12368:
12333:
12292:
12277:
12262:
12247:
12212:
12204:
12181:
12156:
12133:
12112:
12109:
12076:
12050:
12009:
11994:
11979:
11964:
11929:
11921:
11898:
11873:
11850:
11829:
11826:
11811:
11764:
11671:
11670:
11659:
11656:
11641:
11622:
11608:
11607:
11589:
11586:
11571:
11558:
11521:
11422:
11421:
11383:
11363:
11314:
11313:
11275:
11255:
11214:
11213:
11152:
11120:Fresnel integrals
11036:
11035:
11024:
11021:
11008:
10976:
10955:
10954:
10936:
10930:
10916:
10852:
10675:
10645:Lévy distribution
10495:critical behavior
10467:Benoît Mandelbrot
10284:
10118:{\displaystyle c}
9995:
9952:
9924:
9862:
9850:
9826:
9774:
9754:
9689:
9667:
9617:
9613:
9545:
9456:{\displaystyle W}
9422:
9407:
9376:{\displaystyle U}
9179:
9164:
9093:
9028:
9008:
8946:
8718:
8695:
8633:
8501:
8386:
8332:, this leads to:
8198:
7969:
7968:
7811:special functions
7763:Lévy distribution
7209:
7100:
6420:
6419: where
6382:
6376:
6354:
6325:
6301:
6291:
6285:
6251:
6221:
6206:
6104:
6103: where
5875:
5833:
5741:
5740: where
5731:
5656:
5635:
5587:
5493:
5468:Lévy distribution
5444:
5413:
5279:
5229:
5228: where
5217:
5178:
5153:
5147:
5044:
5030:
5029: where
5018:
4994:
4984:
4977:
4914:to arrive at the
4831:
4807:
4675:{\displaystyle x}
4556:
4528:
4508:
4383:
4276:
3988:
3675:
3644:
3574:
3478:
3341:
3153:
3048:
2978:
2962:
2929:
2726:
2693:
2080:
2048:
1716:Fourier transform
1622:Lévy distribution
1243:
1242:
1212:
1185:
1158:
1150:
119:
109:
73:
63:
16611:
16561:
16560:
16551:
16550:
16490:Compound Poisson
16465:
16453:
16422:von Mises–Fisher
16418:
16406:
16394:
16356:Circular uniform
16352:
16272:
16216:
16187:
16148:
16147:
16050:Marchenko–Pastur
15913:Geometric stable
15830:Truncated normal
15723:Inverse Gaussian
15629:Hyperexponential
15468:Beta rectangular
15436:bounded interval
15431:
15430:
15299:Discrete uniform
15284:Poisson binomial
15235:
15234:
15210:
15203:
15196:
15187:
15186:
15181:
15180:
15155:(6): 1187–1228.
15140:
15134:
15133:
15121:
15115:
15114:
15106:
15100:
15099:
15074:(5): 5327–5343.
15063:
15054:
15053:
15036:(5): 2670–2689.
15025:
15016:
15011:
15005:
15004:
14984:
14978:
14977:
14949:
14943:
14942:
14917:(5): 4677–4683.
14906:
14900:
14899:
14879:
14873:
14872:
14854:
14834:
14828:
14827:
14807:
14796:
14790:
14789:
14777:
14767:
14761:
14760:
14743:(354): 340–344.
14732:
14726:
14725:
14707:
14698:
14692:
14691:
14689:
14665:
14659:
14658:
14622:
14613:
14612:
14584:
14578:
14577:
14567:
14558:
14557:
14537:
14531:
14530:
14522:
14516:
14515:
14507:
14501:
14500:
14476:
14470:
14469:
14457:
14442:
14441:
14407:
14387:
14378:
14377:
14349:
14340:
14339:
14303:
14292:
14291:
14265:
14254:
14253:
14251:
14250:
14238:Siegrist, Kyle.
14235:
14229:
14228:
14226:
14225:
14219:
14213:. Archived from
14212:
14203:
14186:
14185:
14155:
14146:
14145:
14117:
14111:
14110:
14082:
14076:
14075:
14047:
14041:
14040:
14024:
14018:
14017:
13989:
13841:
13839:
13838:
13833:
13831:
13827:
13826:
13809:
13805:
13804:
13803:
13794:
13786:
13779:
13778:
13777:
13769:
13764:
13756:
13746:
13742:
13741:
13740:
13731:
13723:
13710:
13708:
13707:
13699:
13694:
13689:
13679:
13678:
13672:
13655:
13651:
13650:
13649:
13640:
13632:
13622:
13621:
13620:
13612:
13607:
13599:
13592:
13588:
13587:
13586:
13577:
13569:
13556:
13554:
13553:
13545:
13540:
13535:
13528:
13527:
13510:
13506:
13487:
13478:
13457:
13453:
13452:
13451:
13439:
13430:
13422:
13421:
13420:
13412:
13407:
13399:
13392:
13388:
13387:
13386:
13374:
13365:
13348:
13346:
13345:
13337:
13332:
13327:
13320:
13319:
13310:
13306:
13287:
13278:
13257:
13253:
13252:
13251:
13239:
13230:
13222:
13221:
13220:
13212:
13207:
13199:
13189:
13185:
13184:
13183:
13171:
13162:
13148:
13146:
13145:
13137:
13132:
13127:
13117:
13116:
13107:
13103:
13084:
13075:
13043:
13041:
13040:
13035:
13024:
13023:
12994:
12992:
12991:
12986:
12984:
12980:
12976:
12975:
12972:
12971:
12962:
12961:
12960:
12951:
12950:
12940:
12931:
12922:
12916:
12907:
12901:
12892:
12886:
12877:
12871:
12862:
12854:
12853:
12844:
12843:
12838:
12835:
12833:
12832:
12831:
12822:
12821:
12811:
12810:
12806:
12797:
12787:
12786:
12773:
12768:
12764:
12763:
12760:
12759:
12750:
12749:
12748:
12739:
12738:
12728:
12719:
12710:
12704:
12695:
12689:
12680:
12674:
12665:
12659:
12650:
12638:
12629:
12621:
12620:
12611:
12610:
12605:
12602:
12600:
12592:
12591:
12582:
12577:
12573:
12572:
12569:
12568:
12559:
12558:
12557:
12548:
12547:
12537:
12528:
12519:
12513:
12504:
12498:
12489:
12483:
12474:
12468:
12459:
12451:
12450:
12441:
12440:
12435:
12432:
12427:
12426:
12422:
12413:
12402:
12393:
12389:
12370:
12361:
12340:
12336:
12335:
12332:
12331:
12322:
12321:
12320:
12311:
12310:
12300:
12294:
12285:
12279:
12270:
12264:
12255:
12249:
12240:
12232:
12231:
12222:
12221:
12216:
12213:
12211:
12210:
12206:
12197:
12187:
12183:
12174:
12163:
12162:
12158:
12149:
12139:
12135:
12126:
12115:
12113:
12111:
12110:
12102:
12100:
12099:
12089:
12088:
12087:
12078:
12077:
12069:
12062:
12057:
12053:
12052:
12049:
12048:
12039:
12038:
12037:
12028:
12027:
12017:
12011:
12002:
11996:
11987:
11981:
11972:
11966:
11957:
11949:
11948:
11939:
11938:
11933:
11930:
11928:
11927:
11923:
11914:
11904:
11900:
11891:
11880:
11879:
11875:
11866:
11856:
11852:
11843:
11832:
11830:
11828:
11827:
11819:
11813:
11812:
11804:
11798:
11789:
11785:
11766:
11757:
11725:
11723:
11722:
11717:
11715:
11714:
11705:
11704:
11699:
11687:
11685:
11684:
11679:
11677:
11673:
11672:
11666:
11662:
11660:
11658:
11657:
11649:
11643:
11642:
11637:
11631:
11624:
11623:
11615:
11609:
11606:
11605:
11596:
11592:
11590:
11588:
11587:
11579:
11573:
11572:
11567:
11561:
11559:
11551:
11546:
11542:
11523:
11514:
11486:
11484:
11483:
11478:
11467:
11466:
11448:
11446:
11445:
11440:
11438:
11434:
11433:
11429:
11428:
11424:
11420:
11412:
11401:
11397:
11384:
11376:
11369:
11365:
11362:
11361:
11353:
11341:
11325:
11321:
11320:
11316:
11312:
11304:
11293:
11289:
11276:
11268:
11261:
11257:
11254:
11253:
11245:
11233:
11215:
11212:
11211:
11206:
11197:
11186:
11182:
11177:
11173:
11154:
11145:
11117:
11115:
11114:
11109:
11088:
11086:
11085:
11080:
11057:
11055:
11054:
11049:
11047:
11043:
11042:
11038:
11037:
11031:
11027:
11025:
11023:
11022:
11017:
11011:
11010:
11009:
11004:
10996:
10986:
10979:
10978:
10977:
10969:
10956:
10953:
10952:
10943:
10939:
10937:
10935:
10931:
10926:
10920:
10919:
10918:
10917:
10912:
10904:
10890:
10877:
10873:
10854:
10845:
10817:
10815:
10814:
10809:
10798:
10797:
10773:
10771:
10770:
10765:
10711:
10709:
10708:
10703:
10677:
10668:
10640:
10638:
10637:
10632:
10576:
10574:
10573:
10568:
10488:
10486:
10485:
10480:
10434:
10432:
10431:
10426:
10408:
10406:
10405:
10400:
10382:
10380:
10379:
10374:
10351:
10350:
10334:
10332:
10331:
10326:
10324:
10323:
10285:
10277:
10218:
10216:
10215:
10210:
10187:
10186:
10164:
10162:
10161:
10156:
10144:
10142:
10141:
10136:
10124:
10122:
10121:
10116:
10104:
10102:
10101:
10096:
10081:
10079:
10078:
10073:
10050:
10049:
10024:
10022:
10021:
10016:
10014:
10013:
9996:
9988:
9953:
9945:
9925:
9920:
9912:
9885:
9883:
9882:
9877:
9872:
9868:
9867:
9863:
9861:
9851:
9843:
9840:
9827:
9819:
9816:
9789:
9785:
9775:
9767:
9755:
9747:
9732:
9730:
9729:
9724:
9704:
9702:
9701:
9696:
9691:
9690:
9685:
9674:
9672:
9668:
9663:
9625:
9618:
9616:
9615:
9614:
9606:
9581:
9549:
9547:
9546:
9544:
9533:
9531:
9527:
9526:
9525:
9491:
9489:
9488:
9483:
9462:
9460:
9459:
9454:
9439:
9437:
9436:
9431:
9429:
9425:
9424:
9415:
9409:
9400:
9382:
9380:
9379:
9374:
9357:
9355:
9354:
9349:
9328:
9326:
9325:
9320:
9309:
9308:
9281:
9279:
9278:
9273:
9234:
9232:
9231:
9226:
9224:
9199:
9198:
9184:
9180:
9172:
9165:
9163:
9155:
9117:
9114:
9109:
9094:
9086:
9078:
9074:
9070:
9048:
9047:
9033:
9029:
9024:
9016:
9009:
9007:
8999:
8998:
8997:
8978:
8975:
8970:
8947:
8939:
8921:
8920:
8900:
8898:
8897:
8892:
8884:
8883:
8864:
8862:
8861:
8856:
8848:
8774:
8772:
8771:
8766:
8764:
8760:
8738:
8737:
8723:
8719:
8717:
8703:
8696:
8694:
8686:
8685:
8684:
8665:
8662:
8657:
8634:
8626:
8578:
8576:
8575:
8570:
8547:
8546:
8524:
8522:
8521:
8516:
8514:
8510:
8502:
8500:
8492:
8491:
8490:
8481:
8480:
8461:
8458:
8453:
8438:
8437:
8409:
8404:
8387:
8379:
8325:
8323:
8322:
8317:
8312:
8308:
8300:
8299:
8277:
8276:
8250:
8249:
8221:
8216:
8199:
8191:
8135:
8097:
8095:
8094:
8089:
8071:
8069:
8068:
8063:
8058:
8026:
8024:
8023:
8018:
8000:
7998:
7997:
7992:
7899:
7897:
7896:
7891:
7849:
7847:
7846:
7841:
7824:
7823:
7804:
7802:
7801:
7796:
7760:
7758:
7757:
7752:
7734:
7732:
7731:
7726:
7721:
7685:
7683:
7682:
7677:
7659:
7657:
7656:
7651:
7630:
7628:
7627:
7622:
7594:
7592:
7591:
7586:
7545:
7543:
7542:
7537:
7498:
7496:
7495:
7490:
7472:
7470:
7469:
7464:
7224:
7222:
7221:
7216:
7214:
7210:
7208:
7206:
7201:
7188:
7183:
7173:
7171:
7166:
7157:
7156:
7143:
7138:
7129:
7128:
7118:
7102:
7101:
7093:
7091:
7087:
7085:
7080:
7067:
7062:
7033:
7032:
7020:
7019:
6989:
6987:
6986:
6981:
6961:
6955:
6953:
6952:
6947:
6945:
6941:
6922:
6921:
6916:
6907:
6906:
6897:
6892:
6891:
6858:
6857:
6852:
6843:
6842:
6833:
6828:
6827:
6812:
6811:
6806:
6797:
6796:
6787:
6779:
6778:
6773:
6764:
6763:
6754:
6746:
6745:
6727:
6726:
6693:
6691:
6690:
6685:
6673:
6671:
6670:
6665:
6642:
6640:
6639:
6634:
6588:
6586:
6585:
6580:
6569:
6568:
6552:
6550:
6549:
6544:
6517:
6515:
6514:
6509:
6498:
6497:
6492:
6491:
6474:
6472:
6471:
6466:
6440:
6438:
6437:
6432:
6421:
6418:
6413:
6412:
6411:
6410:
6405:
6396:
6383:
6381:
6377:
6369:
6357:
6355:
6347:
6335:
6331:
6330:
6326:
6318:
6312:
6311:
6302:
6294:
6292:
6290:
6286:
6278:
6266:
6259:
6255:
6254:
6253:
6252:
6247:
6246:
6238:
6232:
6222:
6214:
6207:
6199:
6196:
6191:
6164:
6162:
6161:
6156:
6151:
6127:
6125:
6124:
6119:
6105:
6102:
6097:
6096:
6095:
6094:
6059:
6058:
6049:
6048:
6032:
6027:
6005:
6003:
6002:
5997:
5995:
5994:
5974:
5972:
5971:
5966:
5948:
5946:
5945:
5940:
5932:
5931:
5915:
5913:
5912:
5907:
5896:
5895:
5877:
5876:
5868:
5866:
5865:
5847:
5845:
5844:
5839:
5834:
5829:
5820:
5818:
5817:
5812:
5801:
5800:
5781:
5779:
5778:
5773:
5758:
5757:
5742:
5739:
5734:
5733:
5732:
5730:
5722:
5721:
5720:
5704:
5694:
5693:
5689:
5676:
5675:
5657:
5655:
5654:
5653:
5649:
5636:
5631:
5622:
5608:
5607:
5589:
5588:
5580:
5578:
5577:
5560:
5558:
5557:
5552:
5533:
5531:
5530:
5525:
5514:
5513:
5495:
5494:
5486:
5484:
5483:
5465:
5463:
5462:
5457:
5446:
5445:
5437:
5424:
5422:
5421:
5416:
5414:
5406:
5388:
5386:
5385:
5380:
5368:
5366:
5365:
5360:
5358:
5357:
5341:
5339:
5338:
5333:
5322:
5321:
5299:
5297:
5296:
5291:
5280:
5277:
5256:
5254:
5253:
5248:
5246:
5245:
5230:
5227:
5222:
5218:
5216:
5215:
5214:
5195:
5189:
5188:
5179:
5177:
5176:
5175:
5156:
5154:
5152:
5148:
5140:
5128:
5114:
5113:
5095:
5094:
5089:
5088:
5064:
5062:
5061:
5056:
5045:
5042:
5031:
5028:
5023:
5019:
5011:
5005:
5004:
4995:
4987:
4985:
4983:
4982:
4978:
4970:
4957:
4943:
4942:
4937:
4936:
4913:
4911:
4910:
4907:{\textstyle x=1}
4905:
4887:
4885:
4884:
4879:
4877:
4876:
4872:
4846:
4844:
4843:
4838:
4836:
4832:
4824:
4818:
4817:
4808:
4800:
4792:has the density
4787:
4785:
4784:
4779:
4768:
4767:
4755:
4754:
4738:
4736:
4735:
4730:
4728:
4727:
4717:
4712:
4681:
4679:
4678:
4673:
4659:
4657:
4656:
4651:
4649:
4632:
4631:
4579:
4578:
4577:
4576:
4557:
4554:
4543:
4538:
4529:
4521:
4513:
4509:
4506:
4491:
4490:
4435:
4434:
4433:
4432:
4398:
4393:
4384:
4376:
4368:
4364:
4360:
4352:
4351:
4350:
4349:
4344:
4335:
4319:
4318:
4302:
4297:
4277:
4269:
4251:
4250:
4230:
4228:
4227:
4222:
4192:
4190:
4189:
4184:
4182:
4178:
4177:
4176:
4171:
4162:
4112:
4110:
4109:
4104:
4096:
4054:
4052:
4051:
4046:
4028:
4026:
4025:
4020:
4018:
4014:
4007:
4006:
4002:
3993:
3989:
3984:
3976:
3925:
3924:
3893:
3891:
3890:
3885:
3867:
3865:
3864:
3859:
3833:
3832:
3831:
3813:
3811:
3810:
3805:
3784:
3782:
3781:
3776:
3750:
3748:
3747:
3742:
3721:
3719:
3718:
3713:
3691:
3689:
3688:
3683:
3681:
3677:
3676:
3671:
3651:
3649:
3645:
3640:
3632:
3590:
3589:
3575:
3573:
3572:
3571:
3560:
3551:
3542:
3518:
3516:
3515:
3510:
3490:
3488:
3487:
3482:
3480:
3479:
3471:
3469:
3465:
3464:
3463:
3458:
3452:
3451:
3442:
3436:
3431:
3397:
3395:
3394:
3389:
3357:
3343:
3334:
3325:has the density
3320:
3318:
3317:
3312:
3301:
3300:
3288:
3287:
3277:
3272:
3232:
3230:
3229:
3224:
3176:The distribution
3172:
3170:
3169:
3164:
3159:
3155:
3149:
3141:
3105:
3103:
3102:
3097:
3079:
3077:
3076:
3071:
3059:
3057:
3056:
3051:
3049:
3044:
3033:
3016:
3014:
3013:
3008:
3006:
3005:
2979:
2971:
2963:
2958:
2947:
2930:
2925:
2914:
2891:
2889:
2888:
2883:
2819:
2817:
2816:
2811:
2799:
2797:
2796:
2791:
2777:
2775:
2774:
2769:
2767:
2766:
2749:
2738:
2727:
2719:
2699:
2695:
2689:
2681:
2668:
2664:
2657:
2656:
2645:
2633:
2602:
2600:
2599:
2594:
2592:
2588:
2587:
2583:
2547:
2546:
2541:
2529:
2455:
2453:
2452:
2447:
2422:Parametrizations
2417:
2415:
2414:
2409:
2386:
2384:
2383:
2378:
2360:
2358:
2357:
2352:
2315:
2313:
2312:
2307:
2262:
2260:
2259:
2254:
2242:
2240:
2239:
2234:
2210:
2208:
2207:
2202:
2176:
2174:
2173:
2168:
2128:
2126:
2125:
2120:
2118:
2117:
2100:
2092:
2081:
2073:
2053:
2049:
2044:
2036:
2003:
1995:
1987:
1985:
1984:
1979:
1977:
1973:
1972:
1968:
1932:
1931:
1926:
1914:
1834:
1832:
1831:
1826:
1814:
1813:
1785:
1780:
1746:
1744:
1743:
1738:
1713:
1711:
1710:
1705:
1678:
1676:
1675:
1670:
1658:
1656:
1655:
1650:
1608:
1597:
1591:
1584:
1574:
1558:
1551:
1540:
1534:
1525:
1516:
1492:
1490:
1489:
1484:
1439:
1437:
1436:
1431:
1410:, and undefined
1409:
1407:
1406:
1401:
1375:
1373:
1372:
1367:
1345:
1343:
1342:
1337:
1313:
1311:
1310:
1305:
1266:random variables
1239:
1237:
1236:
1231:
1229:
1228:
1213:
1210:
1206:
1198:
1187:
1178:
1159:
1156:
1152:
1146:
1138:
1106:
1104:
1103:
1098:
1093:
1092:
1048:
1047:
1042:
1029:
1011:
1010:
981:
979:
978:
973:
928:
926:
925:
920:
918:
917:
899:
898:
871:
870:
852:
850:
849:
844:
799:
797:
796:
791:
789:
788:
776:
756:
755:
746:
745:
727:
726:
708:
706:
705:
700:
682:
680:
679:
674:
672:
671:
665:
664:
655:
654:
636:
635:
599:
597:
596:
591:
563:
561:
560:
555:
527:
525:
524:
519:
487:
485:
484:
479:
448:
446:
445:
440:
409:
407:
406:
401:
342:
340:
339:
334:
313:
311:
310:
305:
278:
276:
275:
270:
252:
250:
249:
244:
187:
185:
184:
179:
166:
164:
163:
158:
117:
115:
107:
105:
104:
99:
87:
85:
71:
69:
61:
59:
58:
53:
41:
39:
29:
25:
16619:
16618:
16614:
16613:
16612:
16610:
16609:
16608:
16574:
16573:
16572:
16567:
16539:
16515:Maximum entropy
16473:
16461:
16449:
16439:
16431:
16414:
16402:
16390:
16345:
16332:
16269:Matrix-valued:
16266:
16212:
16183:
16175:
16164:
16152:
16143:
16133:
16027:
16021:
15938:
15864:
15862:
15856:
15785:Maxwell–Jüttner
15634:Hypoexponential
15540:
15538:
15537:supported on a
15532:
15493:Noncentral beta
15453:Balding–Nichols
15435:
15434:supported on a
15426:
15416:
15319:
15313:
15309:Zipf–Mandelbrot
15239:
15230:
15224:
15214:
15184:
15141:
15137:
15122:
15118:
15107:
15103:
15064:
15057:
15026:
15019:
15012:
15008:
15001:
14985:
14981:
14950:
14946:
14907:
14903:
14896:
14880:
14876:
14835:
14831:
14824:
14805:
14797:
14793:
14786:
14768:
14764:
14733:
14729:
14705:
14699:
14695:
14666:
14662:
14623:
14616:
14601:10.1137/1139025
14585:
14581:
14568:
14561:
14554:
14538:
14534:
14523:
14519:
14508:
14504:
14477:
14473:
14458:
14445:
14388:
14381:
14350:
14343:
14328:
14304:
14295:
14288:
14278:10.1007/b137351
14266:
14257:
14248:
14246:
14236:
14232:
14223:
14221:
14217:
14210:
14206:Nolan, John P.
14204:
14189:
14156:
14149:
14118:
14114:
14083:
14079:
14064:10.2307/1911802
14048:
14044:
14025:
14021:
14006:10.2307/2525289
13990:
13983:
13979:
13905:
13848:
13829:
13828:
13822:
13821:
13810:
13799:
13795:
13785:
13784:
13780:
13768:
13755:
13751:
13747:
13736:
13732:
13722:
13721:
13717:
13703:
13695:
13688:
13684:
13677:
13674:
13673:
13671:
13668:
13667:
13656:
13645:
13641:
13631:
13627:
13623:
13611:
13598:
13597:
13593:
13582:
13578:
13568:
13567:
13563:
13549:
13541:
13534:
13533:
13526:
13519:
13518:
13511:
13476:
13469:
13465:
13459:
13458:
13444:
13440:
13428:
13427:
13423:
13411:
13398:
13397:
13393:
13379:
13375:
13363:
13359:
13355:
13341:
13333:
13326:
13325:
13318:
13311:
13276:
13269:
13265:
13259:
13258:
13244:
13240:
13228:
13227:
13223:
13211:
13198:
13194:
13190:
13176:
13172:
13160:
13159:
13155:
13141:
13133:
13126:
13122:
13115:
13108:
13073:
13066:
13062:
13055:
13053:
13050:
13049:
13013:
13009:
13007:
13004:
13003:
12982:
12981:
12967:
12963:
12956:
12952:
12946:
12942:
12941:
12938:
12920:
12905:
12890:
12875:
12860:
12859:
12855:
12849:
12845:
12839:
12837:
12836:
12827:
12823:
12817:
12813:
12812:
12795:
12791:
12782:
12778:
12774:
12772:
12755:
12751:
12744:
12740:
12734:
12730:
12729:
12726:
12708:
12693:
12678:
12663:
12648:
12627:
12626:
12622:
12616:
12612:
12606:
12604:
12603:
12593:
12587:
12583:
12581:
12564:
12560:
12553:
12549:
12543:
12539:
12538:
12535:
12517:
12502:
12487:
12472:
12457:
12456:
12452:
12446:
12442:
12436:
12434:
12433:
12411:
12407:
12403:
12401:
12394:
12359:
12352:
12348:
12342:
12341:
12327:
12323:
12316:
12312:
12306:
12302:
12301:
12298:
12283:
12268:
12253:
12238:
12237:
12233:
12227:
12223:
12217:
12215:
12214:
12195:
12191:
12172:
12168:
12164:
12147:
12143:
12124:
12120:
12116:
12114:
12101:
12095:
12091:
12090:
12083:
12079:
12068:
12064:
12063:
12061:
12044:
12040:
12033:
12029:
12023:
12019:
12018:
12015:
12000:
11985:
11970:
11955:
11954:
11950:
11944:
11940:
11934:
11932:
11931:
11912:
11908:
11889:
11885:
11881:
11864:
11860:
11841:
11837:
11833:
11831:
11818:
11814:
11803:
11799:
11797:
11790:
11755:
11748:
11744:
11737:
11735:
11732:
11731:
11710:
11706:
11700:
11698:
11697:
11695:
11692:
11691:
11661:
11648:
11644:
11636:
11632:
11630:
11629:
11625:
11614:
11610:
11601:
11597:
11591:
11578:
11574:
11566:
11562:
11560:
11550:
11512:
11505:
11501:
11496:
11493:
11492:
11462:
11458:
11456:
11453:
11452:
11416:
11408:
11395:
11391:
11375:
11374:
11370:
11357:
11349:
11345:
11339:
11335:
11308:
11300:
11287:
11283:
11267:
11266:
11262:
11249:
11241:
11237:
11231:
11227:
11220:
11216:
11207:
11202:
11201:
11193:
11181:
11143:
11136:
11132:
11127:
11124:
11123:
11094:
11091:
11090:
11065:
11062:
11061:
11026:
11016:
11012:
10997:
10995:
10991:
10987:
10985:
10984:
10980:
10968:
10961:
10957:
10948:
10944:
10938:
10925:
10921:
10905:
10903:
10899:
10895:
10891:
10889:
10888:
10884:
10843:
10836:
10832:
10827:
10824:
10823:
10820:Lommel function
10787:
10783:
10781:
10778:
10777:
10723:
10720:
10719:
10666:
10652:
10649:
10648:
10590:
10587:
10586:
10529:
10526:
10525:
10522:
10474:
10471:
10470:
10462:self-similarity
10450:
10439:for generating
10414:
10411:
10410:
10388:
10385:
10384:
10346:
10342:
10340:
10337:
10336:
10319:
10318:
10307:
10276:
10264:
10263:
10252:
10233:
10232:
10224:
10221:
10220:
10182:
10178:
10170:
10167:
10166:
10150:
10147:
10146:
10130:
10127:
10126:
10110:
10107:
10106:
10090:
10087:
10086:
10045:
10041:
10033:
10030:
10029:
10009:
10008:
9997:
9987:
9984:
9983:
9972:
9944:
9937:
9936:
9913:
9911:
9891:
9888:
9887:
9842:
9841:
9818:
9817:
9815:
9811:
9766:
9765:
9761:
9760:
9756:
9746:
9738:
9735:
9734:
9712:
9709:
9708:
9675:
9673:
9626:
9624:
9620:
9619:
9605:
9601:
9582:
9550:
9548:
9537:
9532:
9521:
9517:
9510:
9506:
9505:
9497:
9494:
9493:
9471:
9468:
9467:
9448:
9445:
9444:
9413:
9398:
9394:
9390:
9388:
9385:
9384:
9368:
9365:
9364:
9334:
9331:
9330:
9301:
9297:
9295:
9292:
9291:
9288:
9255:
9252:
9251:
9246:and subsequent
9240:
9222:
9221:
9185:
9171:
9167:
9166:
9156:
9118:
9116:
9110:
9099:
9085:
9076:
9075:
9034:
9017:
9015:
9011:
9010:
9000:
8993:
8989:
8979:
8977:
8971:
8960:
8955:
8951:
8938:
8931:
8916:
8912:
8908:
8906:
8903:
8902:
8879:
8875:
8870:
8867:
8866:
8844:
8815:
8812:
8811:
8724:
8707:
8702:
8698:
8697:
8687:
8680:
8676:
8666:
8664:
8658:
8647:
8642:
8638:
8625:
8584:
8581:
8580:
8542:
8538:
8530:
8527:
8526:
8493:
8486:
8482:
8476:
8472:
8462:
8460:
8454:
8443:
8415:
8411:
8405:
8400:
8395:
8391:
8378:
8337:
8334:
8333:
8272:
8268:
8255:
8251:
8227:
8223:
8217:
8212:
8207:
8203:
8190:
8149:
8146:
8145:
8142:
8122:
8077:
8074:
8073:
8054:
8043:
8040:
8039:
8006:
8003:
8002:
7980:
7977:
7976:
7885:
7882:
7881:
7835:
7832:
7831:
7790:
7787:
7786:
7740:
7737:
7736:
7717:
7706:
7703:
7702:
7665:
7662:
7661:
7639:
7636:
7635:
7616:
7613:
7612:
7574:
7571:
7570:
7516:
7513:
7512:
7478:
7475:
7474:
7443:
7440:
7439:
7424:
7387:
7366:
7357:
7348:
7341:
7323:
7314:
7302:
7295:
7288:
7233:
7212:
7211:
7202:
7197:
7184:
7179:
7174:
7167:
7162:
7152:
7148:
7139:
7134:
7124:
7120:
7119:
7117:
7110:
7104:
7103:
7092:
7081:
7076:
7063:
7058:
7053:
7049:
7048:
7041:
7035:
7034:
7028:
7024:
7015:
7011:
7004:
6997:
6995:
6992:
6991:
6975:
6972:
6971:
6959:
6917:
6912:
6911:
6902:
6898:
6893:
6887:
6883:
6853:
6848:
6847:
6838:
6834:
6829:
6823:
6819:
6807:
6802:
6801:
6792:
6788:
6783:
6774:
6769:
6768:
6759:
6755:
6750:
6741:
6737:
6722:
6718:
6711:
6707:
6699:
6696:
6695:
6679:
6676:
6675:
6659:
6656:
6655:
6622:
6619:
6618:
6595:
6564:
6560:
6558:
6555:
6554:
6523:
6520:
6519:
6493:
6487:
6486:
6485:
6483:
6480:
6479:
6454:
6451:
6450:
6417:
6406:
6401:
6400:
6392:
6388:
6384:
6368:
6361:
6356:
6346:
6317:
6313:
6307:
6303:
6293:
6277:
6270:
6265:
6264:
6260:
6242:
6234:
6233:
6231:
6227:
6223:
6213:
6212:
6208:
6198:
6192:
6187:
6181:
6178:
6177:
6147:
6136:
6133:
6132:
6101:
6090:
6086:
6082:
6078:
6054:
6050:
6038:
6034:
6028:
6023:
6017:
6014:
6013:
5990:
5986:
5984:
5981:
5980:
5954:
5951:
5950:
5927:
5923:
5921:
5918:
5917:
5891:
5887:
5867:
5861:
5860:
5859:
5857:
5854:
5853:
5828:
5826:
5823:
5822:
5796:
5792:
5790:
5787:
5786:
5753:
5749:
5738:
5723:
5716:
5712:
5705:
5703:
5699:
5695:
5685:
5681:
5677:
5671:
5667:
5645:
5641:
5637:
5630:
5626:
5621:
5603:
5599:
5579:
5573:
5572:
5571:
5569:
5566:
5565:
5543:
5540:
5539:
5509:
5505:
5485:
5479:
5478:
5477:
5475:
5472:
5471:
5436:
5432:
5430:
5427:
5426:
5405:
5397:
5394:
5393:
5374:
5371:
5370:
5353:
5349:
5347:
5344:
5343:
5317:
5313:
5308:
5305:
5304:
5278: and
5276:
5262:
5259:
5258:
5241:
5237:
5226:
5210:
5206:
5199:
5194:
5190:
5184:
5180:
5171:
5167:
5160:
5155:
5139:
5132:
5127:
5109:
5105:
5090:
5084:
5083:
5082:
5080:
5077:
5076:
5070:conjugate prior
5043: and
5041:
5027:
5010:
5006:
5000:
4996:
4986:
4969:
4965:
4961:
4956:
4938:
4932:
4931:
4930:
4928:
4925:
4924:
4893:
4890:
4889:
4868:
4864:
4860:
4852:
4849:
4848:
4823:
4819:
4813:
4809:
4799:
4797:
4794:
4793:
4763:
4759:
4750:
4746:
4744:
4741:
4740:
4723:
4719:
4713:
4702:
4690:
4687:
4686:
4667:
4664:
4663:
4647:
4646:
4627:
4623:
4572:
4568:
4553:
4549:
4545:
4539:
4534:
4520:
4511:
4510:
4505:
4486:
4482:
4428:
4424:
4404:
4400:
4394:
4389:
4375:
4366:
4365:
4345:
4340:
4339:
4331:
4324:
4320:
4308:
4304:
4298:
4290:
4285:
4281:
4268:
4261:
4246:
4242:
4238:
4236:
4233:
4232:
4198:
4195:
4194:
4172:
4167:
4166:
4158:
4151:
4147:
4118:
4115:
4114:
4092:
4063:
4060:
4059:
4034:
4031:
4030:
3998:
3994:
3977:
3975:
3971:
3970:
3945:
3941:
3920:
3916:
3914:
3911:
3910:
3873:
3870:
3869:
3847:
3844:
3843:
3840:
3829:
3828:
3826:
3793:
3790:
3789:
3764:
3761:
3760:
3727:
3724:
3723:
3701:
3698:
3697:
3692:where Γ is the
3652:
3650:
3633:
3631:
3627:
3585:
3581:
3580:
3576:
3561:
3556:
3555:
3547:
3546:
3541:
3524:
3521:
3520:
3498:
3495:
3494:
3470:
3459:
3454:
3453:
3447:
3443:
3438:
3432:
3421:
3416:
3412:
3411:
3403:
3400:
3399:
3353:
3332:
3330:
3327:
3326:
3296:
3292:
3283:
3279:
3273:
3262:
3250:
3247:
3246:
3185:
3182:
3181:
3178:
3142:
3139:
3135:
3115:
3112:
3111:
3085:
3082:
3081:
3065:
3062:
3061:
3060:independent of
3034:
3032:
3024:
3021:
3020:
3001:
3000:
2989:
2970:
2948:
2946:
2943:
2942:
2931:
2915:
2913:
2906:
2905:
2897:
2894:
2893:
2844:
2841:
2840:
2805:
2802:
2801:
2785:
2782:
2781:
2762:
2761:
2750:
2745:
2734:
2718:
2712:
2711:
2700:
2682:
2679:
2675:
2646:
2641:
2640:
2629:
2628:
2624:
2617:
2616:
2608:
2605:
2604:
2552:
2548:
2542:
2537:
2536:
2525:
2512:
2508:
2461:
2458:
2457:
2435:
2432:
2431:
2424:
2403:
2400:
2399:
2366:
2363:
2362:
2340:
2337:
2336:
2295:
2292:
2291:
2248:
2245:
2244:
2228:
2225:
2224:
2190:
2187:
2186:
2141:
2138:
2137:
2113:
2112:
2101:
2096:
2088:
2072:
2066:
2065:
2054:
2037:
2035:
2031:
2018:
2017:
2009:
2006:
2005:
2001:
1989:
1937:
1933:
1927:
1922:
1921:
1910:
1897:
1893:
1846:
1843:
1842:
1803:
1799:
1781:
1773:
1752:
1749:
1748:
1723:
1720:
1719:
1690:
1687:
1686:
1664:
1661:
1660:
1644:
1641:
1640:
1610:
1603:
1600:strictly stable
1593:
1586:
1576:
1573:
1566:
1560:
1553:
1546:
1536:
1530:
1528:random variable
1524:
1518:
1515:
1509:
1499:
1466:
1463:
1462:
1459:Vilfredo Pareto
1419:
1416:
1415:
1389:
1386:
1385:
1355:
1352:
1351:
1319:
1316:
1315:
1299:
1296:
1295:
1224:
1223:
1209:
1207:
1202:
1194:
1176:
1170:
1169:
1155:
1153:
1139:
1136:
1123:
1122:
1114:
1111:
1110:
1107:
1088:
1087:
1043:
1038:
1037:
1025:
1006:
1005:
999:
996:
995:
983:
934:
931:
930:
913:
912:
891:
887:
866:
865:
859:
856:
855:
854:
805:
802:
801:
784:
783:
772:
751:
747:
741:
737:
722:
721:
715:
712:
711:
710:
688:
685:
684:
667:
666:
660:
656:
650:
646:
631:
630:
624:
621:
620:
579:
576:
575:
570:Excess kurtosis
543:
540:
539:
507:
504:
503:
467:
464:
463:
428:
425:
424:
389:
386:
385:
319:
316:
315:
293:
290:
289:
258:
255:
254:
232:
229:
228:
201:
199:scale parameter
193:
173:
170:
169:
168:
134:
131:
130:
116:
110:
93:
90:
89:
86:
80:
70:
64:
47:
44:
43:
40:
34:
24:
17:
12:
11:
5:
16617:
16607:
16606:
16601:
16596:
16591:
16586:
16569:
16568:
16566:
16565:
16555:
16544:
16541:
16540:
16538:
16537:
16532:
16527:
16522:
16517:
16512:
16510:Location–scale
16507:
16502:
16497:
16492:
16487:
16481:
16479:
16475:
16474:
16472:
16471:
16466:
16459:
16454:
16446:
16444:
16433:
16432:
16430:
16429:
16424:
16419:
16412:
16407:
16400:
16395:
16388:
16383:
16378:
16373:
16371:Wrapped Cauchy
16368:
16366:Wrapped normal
16363:
16358:
16353:
16342:
16340:
16334:
16333:
16331:
16330:
16329:
16328:
16323:
16321:Normal-inverse
16318:
16313:
16303:
16302:
16301:
16291:
16283:
16278:
16273:
16264:
16263:
16262:
16252:
16244:
16239:
16234:
16229:
16228:
16227:
16217:
16210:
16209:
16208:
16203:
16193:
16188:
16180:
16178:
16170:
16169:
16166:
16165:
16163:
16162:
16156:
16154:
16145:
16139:
16138:
16135:
16134:
16132:
16131:
16126:
16121:
16113:
16105:
16097:
16088:
16079:
16070:
16061:
16052:
16047:
16042:
16037:
16031:
16029:
16023:
16022:
16020:
16019:
16014:
16012:Variance-gamma
16009:
16004:
15996:
15991:
15986:
15981:
15976:
15971:
15963:
15958:
15957:
15956:
15946:
15941:
15936:
15930:
15925:
15920:
15915:
15910:
15905:
15900:
15892:
15887:
15879:
15874:
15868:
15866:
15858:
15857:
15855:
15854:
15852:Wilks's lambda
15849:
15848:
15847:
15837:
15832:
15827:
15822:
15817:
15812:
15807:
15802:
15797:
15792:
15790:Mittag-Leffler
15787:
15782:
15777:
15772:
15767:
15762:
15757:
15752:
15747:
15742:
15737:
15732:
15731:
15730:
15720:
15711:
15706:
15701:
15700:
15699:
15689:
15687:gamma/Gompertz
15684:
15683:
15682:
15677:
15667:
15662:
15657:
15656:
15655:
15643:
15642:
15641:
15636:
15631:
15621:
15620:
15619:
15609:
15604:
15599:
15598:
15597:
15596:
15595:
15585:
15575:
15570:
15565:
15560:
15555:
15550:
15544:
15542:
15539:semi-infinite
15534:
15533:
15531:
15530:
15525:
15520:
15515:
15510:
15505:
15500:
15495:
15490:
15485:
15480:
15475:
15470:
15465:
15460:
15455:
15450:
15445:
15439:
15437:
15428:
15422:
15421:
15418:
15417:
15415:
15414:
15409:
15404:
15399:
15394:
15389:
15384:
15379:
15374:
15369:
15364:
15359:
15354:
15349:
15344:
15339:
15334:
15329:
15323:
15321:
15318:with infinite
15315:
15314:
15312:
15311:
15306:
15301:
15296:
15291:
15286:
15281:
15280:
15279:
15272:Hypergeometric
15269:
15264:
15259:
15254:
15249:
15243:
15241:
15232:
15226:
15225:
15213:
15212:
15205:
15198:
15190:
15183:
15182:
15135:
15116:
15101:
15055:
15017:
15006:
14999:
14979:
14960:(4): 365–373.
14944:
14901:
14894:
14874:
14852:10.1.1.46.3280
14845:(2): 165–171.
14829:
14822:
14791:
14784:
14762:
14727:
14693:
14660:
14633:(3): 367–474.
14614:
14595:(2): 354–362.
14579:
14559:
14552:
14532:
14517:
14502:
14471:
14443:
14398:(21): 210604.
14379:
14360:(4): 759–774.
14341:
14326:
14293:
14286:
14255:
14230:
14187:
14174:10.1086/258792
14168:(5): 421–440.
14147:
14134:10.1086/294633
14128:(4): 420–429.
14112:
14099:10.1086/294632
14093:(4): 394–419.
14077:
14058:(4): 517–543.
14042:
14019:
13980:
13978:
13975:
13974:
13973:
13962:
13948:
13938:
13928:
13923:has a package
13913:
13904:
13901:
13900:
13899:
13894:
13889:
13884:
13883:
13882:
13877:
13872:
13867:
13859:
13854:
13847:
13844:
13843:
13842:
13825:
13820:
13817:
13814:
13811:
13808:
13802:
13798:
13792:
13789:
13783:
13775:
13772:
13767:
13762:
13759:
13754:
13750:
13745:
13739:
13735:
13729:
13726:
13720:
13716:
13713:
13706:
13702:
13698:
13692:
13687:
13682:
13676:
13675:
13670:
13669:
13666:
13663:
13660:
13657:
13654:
13648:
13644:
13638:
13635:
13630:
13626:
13618:
13615:
13610:
13605:
13602:
13596:
13591:
13585:
13581:
13575:
13572:
13566:
13562:
13559:
13552:
13548:
13544:
13538:
13531:
13525:
13524:
13522:
13517:
13514:
13512:
13509:
13505:
13502:
13499:
13496:
13493:
13490:
13484:
13481:
13475:
13472:
13468:
13464:
13461:
13460:
13456:
13450:
13447:
13443:
13436:
13433:
13426:
13418:
13415:
13410:
13405:
13402:
13396:
13391:
13385:
13382:
13378:
13371:
13368:
13362:
13358:
13354:
13351:
13344:
13340:
13336:
13330:
13323:
13317:
13314:
13312:
13309:
13305:
13302:
13299:
13296:
13293:
13290:
13284:
13281:
13275:
13272:
13268:
13264:
13261:
13260:
13256:
13250:
13247:
13243:
13236:
13233:
13226:
13218:
13215:
13210:
13205:
13202:
13197:
13193:
13188:
13182:
13179:
13175:
13168:
13165:
13158:
13154:
13151:
13144:
13140:
13136:
13130:
13125:
13120:
13114:
13111:
13109:
13106:
13102:
13099:
13096:
13093:
13090:
13087:
13081:
13078:
13072:
13069:
13065:
13061:
13058:
13057:
13033:
13030:
13027:
13022:
13019:
13016:
13012:
13000:
12979:
12970:
12966:
12959:
12955:
12949:
12945:
12937:
12934:
12928:
12925:
12919:
12913:
12910:
12904:
12898:
12895:
12889:
12883:
12880:
12874:
12868:
12865:
12858:
12852:
12848:
12842:
12830:
12826:
12820:
12816:
12809:
12803:
12800:
12794:
12790:
12785:
12781:
12777:
12771:
12767:
12758:
12754:
12747:
12743:
12737:
12733:
12725:
12722:
12716:
12713:
12707:
12701:
12698:
12692:
12686:
12683:
12677:
12671:
12668:
12662:
12656:
12653:
12647:
12644:
12641:
12635:
12632:
12625:
12619:
12615:
12609:
12599:
12596:
12590:
12586:
12580:
12576:
12567:
12563:
12556:
12552:
12546:
12542:
12534:
12531:
12525:
12522:
12516:
12510:
12507:
12501:
12495:
12492:
12486:
12480:
12477:
12471:
12465:
12462:
12455:
12449:
12445:
12439:
12430:
12425:
12419:
12416:
12410:
12406:
12400:
12397:
12395:
12392:
12388:
12385:
12382:
12379:
12376:
12373:
12367:
12364:
12358:
12355:
12351:
12347:
12344:
12343:
12339:
12330:
12326:
12319:
12315:
12309:
12305:
12297:
12291:
12288:
12282:
12276:
12273:
12267:
12261:
12258:
12252:
12246:
12243:
12236:
12230:
12226:
12220:
12209:
12203:
12200:
12194:
12190:
12186:
12180:
12177:
12171:
12167:
12161:
12155:
12152:
12146:
12142:
12138:
12132:
12129:
12123:
12119:
12108:
12105:
12098:
12094:
12086:
12082:
12075:
12072:
12067:
12060:
12056:
12047:
12043:
12036:
12032:
12026:
12022:
12014:
12008:
12005:
11999:
11993:
11990:
11984:
11978:
11975:
11969:
11963:
11960:
11953:
11947:
11943:
11937:
11926:
11920:
11917:
11911:
11907:
11903:
11897:
11894:
11888:
11884:
11878:
11872:
11869:
11863:
11859:
11855:
11849:
11846:
11840:
11836:
11825:
11822:
11817:
11810:
11807:
11802:
11796:
11793:
11791:
11788:
11784:
11781:
11778:
11775:
11772:
11769:
11763:
11760:
11754:
11751:
11747:
11743:
11740:
11739:
11713:
11709:
11703:
11688:
11676:
11669:
11665:
11655:
11652:
11647:
11640:
11635:
11628:
11621:
11618:
11613:
11604:
11600:
11595:
11585:
11582:
11577:
11570:
11565:
11557:
11554:
11549:
11545:
11541:
11538:
11535:
11532:
11529:
11526:
11520:
11517:
11511:
11508:
11504:
11500:
11476:
11473:
11470:
11465:
11461:
11449:
11437:
11432:
11427:
11419:
11415:
11411:
11407:
11404:
11400:
11394:
11390:
11387:
11382:
11379:
11373:
11368:
11360:
11356:
11352:
11348:
11344:
11338:
11334:
11331:
11328:
11324:
11319:
11311:
11307:
11303:
11299:
11296:
11292:
11286:
11282:
11279:
11274:
11271:
11265:
11260:
11252:
11248:
11244:
11240:
11236:
11230:
11226:
11223:
11219:
11210:
11205:
11200:
11196:
11192:
11189:
11185:
11180:
11176:
11172:
11169:
11166:
11163:
11160:
11157:
11151:
11148:
11142:
11139:
11135:
11131:
11107:
11104:
11101:
11098:
11078:
11075:
11072:
11069:
11058:
11046:
11041:
11034:
11030:
11020:
11015:
11007:
11003:
11000:
10994:
10990:
10983:
10975:
10972:
10967:
10964:
10960:
10951:
10947:
10942:
10934:
10929:
10924:
10915:
10911:
10908:
10902:
10898:
10894:
10887:
10883:
10880:
10876:
10872:
10869:
10866:
10863:
10860:
10857:
10851:
10848:
10842:
10839:
10835:
10831:
10807:
10804:
10801:
10796:
10793:
10790:
10786:
10774:
10763:
10760:
10757:
10754:
10751:
10748:
10745:
10742:
10739:
10736:
10733:
10730:
10727:
10712:
10701:
10698:
10695:
10692:
10689:
10686:
10683:
10680:
10674:
10671:
10665:
10662:
10659:
10656:
10641:
10630:
10627:
10624:
10621:
10618:
10615:
10612:
10609:
10606:
10603:
10600:
10597:
10594:
10566:
10563:
10560:
10557:
10554:
10551:
10548:
10545:
10542:
10539:
10536:
10533:
10521:
10518:
10489:equal to 1.7.
10478:
10449:
10446:
10424:
10421:
10418:
10398:
10395:
10392:
10372:
10369:
10366:
10363:
10360:
10357:
10354:
10349:
10345:
10322:
10317:
10314:
10311:
10308:
10306:
10303:
10300:
10297:
10294:
10291:
10288:
10283:
10280:
10275:
10272:
10269:
10266:
10265:
10262:
10259:
10256:
10253:
10251:
10248:
10245:
10242:
10239:
10238:
10236:
10231:
10228:
10208:
10205:
10202:
10199:
10196:
10193:
10190:
10185:
10181:
10177:
10174:
10154:
10134:
10114:
10094:
10071:
10068:
10065:
10062:
10059:
10056:
10053:
10048:
10044:
10040:
10037:
10026:
10025:
10012:
10007:
10004:
10001:
9998:
9994:
9991:
9986:
9985:
9982:
9979:
9976:
9973:
9971:
9968:
9965:
9962:
9959:
9956:
9951:
9948:
9943:
9942:
9940:
9935:
9932:
9928:
9923:
9919:
9916:
9910:
9907:
9904:
9901:
9898:
9895:
9875:
9871:
9866:
9860:
9857:
9854:
9849:
9846:
9839:
9836:
9833:
9830:
9825:
9822:
9814:
9810:
9807:
9804:
9801:
9798:
9795:
9792:
9788:
9784:
9781:
9778:
9773:
9770:
9764:
9759:
9753:
9750:
9745:
9742:
9722:
9719:
9716:
9705:
9694:
9688:
9684:
9681:
9678:
9671:
9666:
9662:
9659:
9656:
9653:
9650:
9647:
9644:
9641:
9638:
9635:
9632:
9629:
9623:
9612:
9609:
9604:
9600:
9597:
9594:
9591:
9588:
9585:
9580:
9577:
9574:
9571:
9568:
9565:
9562:
9559:
9556:
9553:
9543:
9540:
9536:
9530:
9524:
9520:
9516:
9513:
9509:
9504:
9501:
9481:
9478:
9475:
9464:
9452:
9428:
9421:
9418:
9412:
9406:
9403:
9397:
9393:
9372:
9347:
9344:
9341:
9338:
9318:
9315:
9312:
9307:
9304:
9300:
9287:
9284:
9271:
9268:
9265:
9262:
9259:
9239:
9236:
9220:
9217:
9214:
9211:
9208:
9205:
9202:
9197:
9194:
9191:
9188:
9183:
9178:
9175:
9170:
9162:
9159:
9154:
9151:
9148:
9145:
9142:
9139:
9136:
9133:
9130:
9127:
9124:
9121:
9113:
9108:
9105:
9102:
9098:
9092:
9089:
9084:
9081:
9079:
9077:
9073:
9069:
9066:
9063:
9060:
9057:
9054:
9051:
9046:
9043:
9040:
9037:
9032:
9027:
9023:
9020:
9014:
9006:
9003:
8996:
8992:
8988:
8985:
8982:
8974:
8969:
8966:
8963:
8959:
8954:
8950:
8945:
8942:
8937:
8934:
8932:
8930:
8927:
8924:
8919:
8915:
8911:
8910:
8890:
8887:
8882:
8878:
8874:
8854:
8851:
8847:
8843:
8840:
8837:
8834:
8831:
8828:
8825:
8822:
8819:
8789:delta function
8763:
8759:
8756:
8753:
8750:
8747:
8744:
8741:
8736:
8733:
8730:
8727:
8722:
8716:
8713:
8710:
8706:
8701:
8693:
8690:
8683:
8679:
8675:
8672:
8669:
8661:
8656:
8653:
8650:
8646:
8641:
8637:
8632:
8629:
8624:
8621:
8618:
8615:
8612:
8609:
8606:
8603:
8600:
8597:
8594:
8591:
8588:
8568:
8565:
8562:
8559:
8556:
8553:
8550:
8545:
8541:
8537:
8534:
8513:
8509:
8506:
8499:
8496:
8489:
8485:
8479:
8475:
8471:
8468:
8465:
8457:
8452:
8449:
8446:
8442:
8436:
8433:
8430:
8427:
8424:
8421:
8418:
8414:
8408:
8403:
8399:
8394:
8390:
8385:
8382:
8377:
8374:
8371:
8368:
8365:
8362:
8359:
8356:
8353:
8350:
8347:
8344:
8341:
8315:
8311:
8307:
8304:
8298:
8295:
8292:
8289:
8286:
8283:
8280:
8275:
8271:
8267:
8264:
8261:
8258:
8254:
8248:
8245:
8242:
8239:
8236:
8233:
8230:
8226:
8220:
8215:
8211:
8206:
8202:
8197:
8194:
8189:
8186:
8183:
8180:
8177:
8174:
8171:
8168:
8165:
8162:
8159:
8156:
8153:
8141:
8138:
8112:
8111:
8087:
8084:
8081:
8061:
8057:
8053:
8050:
8047:
8036:
8016:
8013:
8010:
7990:
7987:
7984:
7967:
7966:
7963:
7961:
7954:
7951:
7944:
7941:
7937:
7936:
7929:
7922:
7919:
7912:
7909:
7906:
7903:
7900:
7889:
7878:
7877:
7874:
7871:
7868:
7865:
7862:
7859:
7856:
7854:
7851:
7850:
7839:
7829:
7827:
7794:
7775:
7774:
7750:
7747:
7744:
7724:
7720:
7716:
7713:
7710:
7699:
7675:
7672:
7669:
7649:
7646:
7643:
7632:
7631:has no effect.
7620:
7599:with variance
7584:
7581:
7578:
7535:
7532:
7529:
7526:
7523:
7520:
7488:
7485:
7482:
7462:
7459:
7456:
7453:
7450:
7447:
7423:
7420:
7408:
7407:
7383:
7374:
7373:
7372:
7371:
7362:
7353:
7346:
7337:
7329:
7328:
7319:
7310:
7300:
7293:
7286:
7267:'s 1954 book.
7232:
7229:
7205:
7200:
7196:
7192:
7187:
7182:
7178:
7170:
7165:
7161:
7155:
7151:
7147:
7142:
7137:
7133:
7127:
7123:
7116:
7113:
7111:
7109:
7106:
7105:
7099:
7096:
7090:
7084:
7079:
7075:
7071:
7066:
7061:
7057:
7052:
7047:
7044:
7042:
7040:
7037:
7036:
7031:
7027:
7023:
7018:
7014:
7010:
7007:
7005:
7003:
7000:
6999:
6979:
6944:
6940:
6937:
6934:
6931:
6928:
6925:
6920:
6915:
6910:
6905:
6901:
6896:
6890:
6886:
6882:
6879:
6876:
6873:
6870:
6867:
6864:
6861:
6856:
6851:
6846:
6841:
6837:
6832:
6826:
6822:
6818:
6815:
6810:
6805:
6800:
6795:
6791:
6786:
6782:
6777:
6772:
6767:
6762:
6758:
6753:
6749:
6744:
6740:
6736:
6733:
6730:
6725:
6721:
6717:
6714:
6710:
6706:
6703:
6683:
6663:
6652:
6651:
6644:
6632:
6629:
6626:
6603:
6594:
6591:
6578:
6575:
6572:
6567:
6563:
6553:-th moment of
6542:
6539:
6536:
6533:
6530:
6527:
6507:
6504:
6501:
6496:
6490:
6464:
6461:
6458:
6442:
6441:
6430:
6427:
6424:
6416:
6409:
6404:
6399:
6395:
6391:
6387:
6380:
6375:
6372:
6367:
6364:
6360:
6353:
6350:
6345:
6342:
6339:
6334:
6329:
6324:
6321:
6316:
6310:
6306:
6300:
6297:
6289:
6284:
6281:
6276:
6273:
6269:
6263:
6258:
6250:
6245:
6241:
6237:
6230:
6226:
6220:
6217:
6211:
6205:
6202:
6195:
6190:
6186:
6169:of a standard
6154:
6150:
6146:
6143:
6140:
6129:
6128:
6117:
6114:
6111:
6108:
6100:
6093:
6089:
6085:
6081:
6077:
6074:
6071:
6068:
6065:
6062:
6057:
6053:
6047:
6044:
6041:
6037:
6031:
6026:
6022:
5993:
5989:
5964:
5961:
5958:
5938:
5935:
5930:
5926:
5905:
5902:
5899:
5894:
5890:
5886:
5883:
5880:
5874:
5871:
5864:
5837:
5832:
5810:
5807:
5804:
5799:
5795:
5783:
5782:
5771:
5768:
5765:
5761:
5756:
5752:
5748:
5745:
5737:
5729:
5726:
5719:
5715:
5711:
5708:
5702:
5698:
5692:
5688:
5684:
5680:
5674:
5670:
5666:
5663:
5660:
5652:
5648:
5644:
5640:
5634:
5629:
5625:
5620:
5617:
5614:
5611:
5606:
5602:
5598:
5595:
5592:
5586:
5583:
5576:
5550:
5547:
5523:
5520:
5517:
5512:
5508:
5504:
5501:
5498:
5492:
5489:
5482:
5455:
5452:
5449:
5443:
5440:
5435:
5412:
5409:
5404:
5401:
5378:
5356:
5352:
5331:
5328:
5325:
5320:
5316:
5312:
5301:
5300:
5289:
5286:
5283:
5275:
5272:
5269:
5266:
5244:
5240:
5236:
5233:
5225:
5221:
5213:
5209:
5205:
5202:
5198:
5193:
5187:
5183:
5174:
5170:
5166:
5163:
5159:
5151:
5146:
5143:
5138:
5135:
5131:
5126:
5123:
5120:
5117:
5112:
5108:
5104:
5101:
5098:
5093:
5087:
5066:
5065:
5054:
5051:
5048:
5040:
5037:
5034:
5026:
5022:
5017:
5014:
5009:
5003:
4999:
4993:
4990:
4981:
4976:
4973:
4968:
4964:
4960:
4955:
4952:
4949:
4946:
4941:
4935:
4903:
4900:
4897:
4875:
4871:
4867:
4863:
4859:
4856:
4835:
4830:
4827:
4822:
4816:
4812:
4806:
4803:
4777:
4774:
4771:
4766:
4762:
4758:
4753:
4749:
4726:
4722:
4716:
4711:
4708:
4705:
4701:
4697:
4694:
4671:
4645:
4642:
4639:
4635:
4630:
4626:
4621:
4618:
4615:
4612:
4609:
4606:
4603:
4600:
4597:
4594:
4591:
4588:
4585:
4582:
4575:
4571:
4566:
4563:
4560:
4552:
4548:
4542:
4537:
4533:
4527:
4524:
4519:
4516:
4514:
4512:
4507: or
4504:
4501:
4498:
4494:
4489:
4485:
4480:
4477:
4474:
4471:
4468:
4465:
4462:
4459:
4456:
4453:
4450:
4447:
4444:
4441:
4438:
4431:
4427:
4422:
4419:
4416:
4413:
4410:
4407:
4403:
4397:
4392:
4388:
4382:
4379:
4374:
4371:
4369:
4367:
4363:
4359:
4356:
4348:
4343:
4338:
4334:
4330:
4327:
4323:
4317:
4314:
4311:
4307:
4301:
4296:
4293:
4289:
4284:
4280:
4275:
4272:
4267:
4264:
4262:
4260:
4257:
4254:
4249:
4245:
4241:
4240:
4220:
4217:
4214:
4211:
4208:
4205:
4202:
4181:
4175:
4170:
4165:
4161:
4157:
4154:
4150:
4146:
4143:
4140:
4137:
4134:
4131:
4128:
4125:
4122:
4102:
4099:
4095:
4091:
4088:
4085:
4082:
4079:
4076:
4073:
4070:
4067:
4056:
4055:
4044:
4041:
4038:
4017:
4013:
4010:
4005:
4001:
3997:
3992:
3987:
3983:
3980:
3974:
3969:
3966:
3963:
3960:
3957:
3954:
3951:
3948:
3944:
3940:
3937:
3934:
3931:
3928:
3923:
3919:
3883:
3880:
3877:
3857:
3854:
3851:
3839:
3836:
3803:
3800:
3797:
3774:
3771:
3768:
3740:
3737:
3734:
3731:
3711:
3708:
3705:
3694:Gamma function
3680:
3674:
3670:
3667:
3664:
3661:
3658:
3655:
3648:
3643:
3639:
3636:
3630:
3626:
3623:
3620:
3617:
3614:
3611:
3608:
3605:
3602:
3599:
3596:
3593:
3588:
3584:
3579:
3570:
3567:
3564:
3559:
3554:
3550:
3545:
3540:
3537:
3534:
3531:
3528:
3508:
3505:
3502:
3477:
3474:
3468:
3462:
3457:
3450:
3446:
3441:
3435:
3430:
3427:
3424:
3420:
3415:
3410:
3407:
3387:
3384:
3381:
3378:
3375:
3372:
3369:
3366:
3363:
3360:
3356:
3352:
3349:
3346:
3340:
3337:
3310:
3307:
3304:
3299:
3295:
3291:
3286:
3282:
3276:
3271:
3268:
3265:
3261:
3257:
3254:
3222:
3219:
3216:
3213:
3210:
3207:
3204:
3201:
3198:
3195:
3192:
3189:
3177:
3174:
3162:
3158:
3152:
3148:
3145:
3138:
3134:
3131:
3128:
3125:
3122:
3119:
3095:
3092:
3089:
3069:
3047:
3043:
3040:
3037:
3031:
3028:
3004:
2999:
2996:
2993:
2990:
2988:
2985:
2982:
2977:
2974:
2969:
2966:
2961:
2957:
2954:
2951:
2945:
2944:
2941:
2938:
2935:
2932:
2928:
2924:
2921:
2918:
2912:
2911:
2909:
2904:
2901:
2881:
2878:
2875:
2872:
2869:
2866:
2863:
2860:
2857:
2854:
2851:
2848:
2809:
2789:
2780:The ranges of
2765:
2760:
2757:
2754:
2751:
2748:
2744:
2741:
2737:
2733:
2730:
2725:
2722:
2717:
2714:
2713:
2710:
2707:
2704:
2701:
2698:
2692:
2688:
2685:
2678:
2674:
2671:
2667:
2663:
2660:
2655:
2652:
2649:
2644:
2639:
2636:
2632:
2627:
2623:
2622:
2620:
2615:
2612:
2591:
2586:
2582:
2579:
2576:
2573:
2570:
2567:
2564:
2561:
2558:
2555:
2551:
2545:
2540:
2535:
2532:
2528:
2524:
2521:
2518:
2515:
2511:
2507:
2504:
2501:
2498:
2495:
2492:
2489:
2486:
2483:
2480:
2477:
2474:
2471:
2468:
2465:
2445:
2442:
2439:
2423:
2420:
2407:
2394:The parameter
2376:
2373:
2370:
2350:
2347:
2344:
2305:
2302:
2299:
2252:
2232:
2217:central moment
2200:
2197:
2194:
2166:
2163:
2160:
2157:
2154:
2151:
2148:
2145:
2116:
2111:
2108:
2105:
2102:
2099:
2095:
2091:
2087:
2084:
2079:
2076:
2071:
2068:
2067:
2064:
2061:
2058:
2055:
2052:
2047:
2043:
2040:
2034:
2030:
2027:
2024:
2023:
2021:
2016:
2013:
1976:
1971:
1967:
1964:
1961:
1958:
1955:
1952:
1949:
1946:
1943:
1940:
1936:
1930:
1925:
1920:
1917:
1913:
1909:
1906:
1903:
1900:
1896:
1892:
1889:
1886:
1883:
1880:
1877:
1874:
1871:
1868:
1865:
1862:
1859:
1856:
1853:
1850:
1824:
1821:
1818:
1812:
1809:
1806:
1802:
1798:
1795:
1792:
1789:
1784:
1779:
1776:
1772:
1768:
1765:
1762:
1759:
1756:
1736:
1733:
1730:
1727:
1703:
1700:
1697:
1694:
1668:
1648:
1571:
1564:
1541:is said to be
1522:
1513:
1507:
1498:
1495:
1482:
1479:
1476:
1473:
1470:
1429:
1426:
1423:
1399:
1396:
1393:
1365:
1362:
1359:
1335:
1332:
1329:
1326:
1323:
1303:
1253:is said to be
1241:
1240:
1227:
1222:
1219:
1216:
1208:
1205:
1201:
1197:
1193:
1190:
1184:
1181:
1175:
1172:
1171:
1168:
1165:
1162:
1154:
1149:
1145:
1142:
1135:
1132:
1129:
1128:
1126:
1121:
1118:
1096:
1091:
1085:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
1061:
1058:
1055:
1052:
1046:
1041:
1036:
1032:
1028:
1024:
1021:
1018:
1015:
1009:
1003:
992:
986:
985:
971:
968:
965:
962:
959:
956:
953:
950:
947:
944:
941:
938:
916:
911:
908:
905:
902:
897:
894:
890:
886:
883:
880:
877:
874:
869:
863:
842:
839:
836:
833:
830:
827:
824:
821:
818:
815:
812:
809:
787:
782:
779:
775:
771:
768:
765:
762:
759:
754:
750:
744:
740:
736:
733:
730:
725:
719:
698:
695:
692:
670:
663:
659:
653:
649:
645:
642:
639:
634:
628:
618:
612:
611:
608:
602:
601:
589:
586:
583:
572:
566:
565:
553:
550:
547:
536:
530:
529:
517:
514:
511:
496:
490:
489:
477:
474:
471:
457:
451:
450:
438:
435:
432:
418:
412:
411:
399:
396:
393:
379:
373:
372:
369:
363:
362:
359:
353:
352:
332:
329:
326:
323:
303:
300:
297:
268:
265:
262:
242:
239:
236:
217:
211:
210:
177:
156:
153:
150:
147:
144:
141:
138:
127:
121:
120:
97:
78:
75:
74:
51:
32:
15:
9:
6:
4:
3:
2:
16616:
16605:
16602:
16600:
16597:
16595:
16592:
16590:
16587:
16585:
16582:
16581:
16579:
16564:
16556:
16554:
16546:
16545:
16542:
16536:
16533:
16531:
16528:
16526:
16523:
16521:
16518:
16516:
16513:
16511:
16508:
16506:
16503:
16501:
16498:
16496:
16493:
16491:
16488:
16486:
16483:
16482:
16480:
16476:
16470:
16467:
16464:
16460:
16458:
16455:
16452:
16448:
16447:
16445:
16443:
16438:
16434:
16428:
16425:
16423:
16420:
16417:
16413:
16411:
16408:
16405:
16401:
16399:
16396:
16393:
16389:
16387:
16384:
16382:
16379:
16377:
16374:
16372:
16369:
16367:
16364:
16362:
16359:
16357:
16354:
16351:
16350:
16344:
16343:
16341:
16339:
16335:
16327:
16324:
16322:
16319:
16317:
16314:
16312:
16309:
16308:
16307:
16304:
16300:
16297:
16296:
16295:
16292:
16290:
16289:
16284:
16282:
16281:Matrix normal
16279:
16277:
16274:
16271:
16270:
16265:
16261:
16258:
16257:
16256:
16253:
16251:
16250:
16247:Multivariate
16245:
16243:
16240:
16238:
16235:
16233:
16230:
16226:
16223:
16222:
16221:
16218:
16215:
16211:
16207:
16204:
16202:
16199:
16198:
16197:
16194:
16192:
16189:
16186:
16182:
16181:
16179:
16177:
16174:Multivariate
16171:
16161:
16158:
16157:
16155:
16149:
16146:
16140:
16130:
16127:
16125:
16122:
16120:
16118:
16114:
16112:
16110:
16106:
16104:
16102:
16098:
16096:
16094:
16089:
16087:
16085:
16080:
16078:
16076:
16071:
16069:
16067:
16062:
16060:
16058:
16053:
16051:
16048:
16046:
16043:
16041:
16038:
16036:
16033:
16032:
16030:
16026:with support
16024:
16018:
16015:
16013:
16010:
16008:
16005:
16003:
16002:
15997:
15995:
15992:
15990:
15987:
15985:
15982:
15980:
15977:
15975:
15972:
15970:
15969:
15964:
15962:
15959:
15955:
15952:
15951:
15950:
15947:
15945:
15942:
15940:
15939:
15931:
15929:
15926:
15924:
15921:
15919:
15916:
15914:
15911:
15909:
15906:
15904:
15901:
15899:
15898:
15893:
15891:
15888:
15886:
15885:
15880:
15878:
15875:
15873:
15870:
15869:
15867:
15863:on the whole
15859:
15853:
15850:
15846:
15843:
15842:
15841:
15838:
15836:
15835:type-2 Gumbel
15833:
15831:
15828:
15826:
15823:
15821:
15818:
15816:
15813:
15811:
15808:
15806:
15803:
15801:
15798:
15796:
15793:
15791:
15788:
15786:
15783:
15781:
15778:
15776:
15773:
15771:
15768:
15766:
15763:
15761:
15758:
15756:
15753:
15751:
15748:
15746:
15743:
15741:
15738:
15736:
15733:
15729:
15726:
15725:
15724:
15721:
15719:
15717:
15712:
15710:
15707:
15705:
15704:Half-logistic
15702:
15698:
15695:
15694:
15693:
15690:
15688:
15685:
15681:
15678:
15676:
15673:
15672:
15671:
15668:
15666:
15663:
15661:
15660:Folded normal
15658:
15654:
15651:
15650:
15649:
15648:
15644:
15640:
15637:
15635:
15632:
15630:
15627:
15626:
15625:
15622:
15618:
15615:
15614:
15613:
15610:
15608:
15605:
15603:
15600:
15594:
15591:
15590:
15589:
15586:
15584:
15581:
15580:
15579:
15576:
15574:
15571:
15569:
15566:
15564:
15561:
15559:
15556:
15554:
15551:
15549:
15546:
15545:
15543:
15535:
15529:
15526:
15524:
15521:
15519:
15516:
15514:
15511:
15509:
15506:
15504:
15503:Raised cosine
15501:
15499:
15496:
15494:
15491:
15489:
15486:
15484:
15481:
15479:
15476:
15474:
15471:
15469:
15466:
15464:
15461:
15459:
15456:
15454:
15451:
15449:
15446:
15444:
15441:
15440:
15438:
15432:
15429:
15423:
15413:
15410:
15408:
15405:
15403:
15400:
15398:
15395:
15393:
15390:
15388:
15385:
15383:
15380:
15378:
15377:Mixed Poisson
15375:
15373:
15370:
15368:
15365:
15363:
15360:
15358:
15355:
15353:
15350:
15348:
15345:
15343:
15340:
15338:
15335:
15333:
15330:
15328:
15325:
15324:
15322:
15316:
15310:
15307:
15305:
15302:
15300:
15297:
15295:
15292:
15290:
15287:
15285:
15282:
15278:
15275:
15274:
15273:
15270:
15268:
15265:
15263:
15260:
15258:
15257:Beta-binomial
15255:
15253:
15250:
15248:
15245:
15244:
15242:
15236:
15233:
15227:
15222:
15218:
15211:
15206:
15204:
15199:
15197:
15192:
15191:
15188:
15178:
15174:
15170:
15166:
15162:
15158:
15154:
15150:
15146:
15139:
15131:
15127:
15120:
15112:
15105:
15097:
15093:
15089:
15085:
15081:
15077:
15073:
15069:
15062:
15060:
15051:
15047:
15043:
15039:
15035:
15031:
15024:
15022:
15015:
15010:
15002:
14996:
14992:
14991:
14983:
14975:
14971:
14967:
14963:
14959:
14955:
14948:
14940:
14936:
14932:
14928:
14924:
14920:
14916:
14912:
14905:
14897:
14895:9780824788827
14891:
14888:. CRC Press.
14887:
14886:
14878:
14870:
14866:
14862:
14858:
14853:
14848:
14844:
14840:
14833:
14825:
14819:
14815:
14811:
14804:
14803:
14795:
14787:
14781:
14776:
14775:
14766:
14758:
14754:
14750:
14746:
14742:
14738:
14731:
14723:
14719:
14716:: 1109–1136.
14715:
14711:
14704:
14697:
14688:
14683:
14679:
14675:
14671:
14664:
14656:
14652:
14648:
14644:
14640:
14636:
14632:
14628:
14621:
14619:
14610:
14606:
14602:
14598:
14594:
14590:
14583:
14575:
14574:
14566:
14564:
14555:
14553:9780412051715
14549:
14546:. CRC Press.
14545:
14544:
14536:
14528:
14521:
14513:
14506:
14498:
14494:
14490:
14486:
14482:
14475:
14467:
14463:
14456:
14454:
14452:
14450:
14448:
14439:
14435:
14431:
14427:
14423:
14419:
14415:
14411:
14406:
14401:
14397:
14393:
14386:
14384:
14375:
14371:
14367:
14363:
14359:
14355:
14348:
14346:
14337:
14333:
14329:
14323:
14319:
14315:
14311:
14310:
14302:
14300:
14298:
14289:
14283:
14279:
14275:
14271:
14264:
14262:
14260:
14245:
14241:
14234:
14220:on 2011-07-17
14216:
14209:
14202:
14200:
14198:
14196:
14194:
14192:
14183:
14179:
14175:
14171:
14167:
14163:
14162:
14154:
14152:
14143:
14139:
14135:
14131:
14127:
14123:
14116:
14108:
14104:
14100:
14096:
14092:
14088:
14081:
14073:
14069:
14065:
14061:
14057:
14053:
14046:
14038:
14034:
14030:
14023:
14015:
14011:
14007:
14003:
14000:(2): 79–106.
13999:
13995:
13988:
13986:
13981:
13970:
13966:
13963:
13960:
13956:
13952:
13949:
13946:
13942:
13939:
13936:
13932:
13929:
13926:
13922:
13918:
13914:
13911:
13907:
13906:
13898:
13895:
13893:
13890:
13888:
13885:
13881:
13878:
13876:
13873:
13871:
13868:
13866:
13863:
13862:
13860:
13858:
13855:
13853:
13850:
13849:
13818:
13815:
13812:
13806:
13800:
13796:
13790:
13787:
13781:
13773:
13770:
13765:
13760:
13757:
13752:
13748:
13743:
13737:
13733:
13727:
13724:
13718:
13714:
13711:
13700:
13690:
13685:
13680:
13664:
13661:
13658:
13652:
13646:
13642:
13636:
13633:
13628:
13624:
13616:
13613:
13608:
13603:
13600:
13594:
13589:
13583:
13579:
13573:
13570:
13564:
13560:
13557:
13546:
13536:
13529:
13520:
13515:
13513:
13507:
13503:
13500:
13497:
13494:
13491:
13488:
13482:
13479:
13473:
13470:
13466:
13462:
13454:
13448:
13445:
13441:
13434:
13431:
13424:
13416:
13413:
13408:
13403:
13400:
13394:
13389:
13383:
13380:
13376:
13369:
13366:
13360:
13356:
13352:
13349:
13338:
13328:
13321:
13315:
13313:
13307:
13303:
13300:
13297:
13294:
13291:
13288:
13282:
13279:
13273:
13270:
13266:
13262:
13254:
13248:
13245:
13241:
13234:
13231:
13224:
13216:
13213:
13208:
13203:
13200:
13195:
13191:
13186:
13180:
13177:
13173:
13166:
13163:
13156:
13152:
13149:
13138:
13128:
13123:
13118:
13112:
13110:
13104:
13100:
13097:
13094:
13091:
13088:
13085:
13079:
13076:
13070:
13067:
13063:
13059:
13047:
13028:
13020:
13017:
13014:
13010:
13001:
12998:
12977:
12968:
12964:
12957:
12953:
12947:
12943:
12935:
12932:
12926:
12923:
12917:
12911:
12908:
12902:
12896:
12893:
12887:
12881:
12878:
12872:
12866:
12863:
12856:
12850:
12846:
12840:
12828:
12824:
12818:
12814:
12807:
12801:
12798:
12792:
12783:
12779:
12775:
12769:
12765:
12756:
12752:
12745:
12741:
12735:
12731:
12723:
12720:
12714:
12711:
12705:
12699:
12696:
12690:
12684:
12681:
12675:
12669:
12666:
12660:
12654:
12651:
12645:
12642:
12639:
12633:
12630:
12623:
12617:
12613:
12607:
12597:
12594:
12588:
12584:
12578:
12574:
12565:
12561:
12554:
12550:
12544:
12540:
12532:
12529:
12523:
12520:
12514:
12508:
12505:
12499:
12493:
12490:
12484:
12478:
12475:
12469:
12463:
12460:
12453:
12447:
12443:
12437:
12428:
12423:
12417:
12414:
12408:
12398:
12396:
12390:
12386:
12383:
12380:
12377:
12374:
12371:
12365:
12362:
12356:
12353:
12349:
12345:
12337:
12328:
12324:
12317:
12313:
12307:
12303:
12295:
12289:
12286:
12280:
12274:
12271:
12265:
12259:
12256:
12250:
12244:
12241:
12234:
12228:
12224:
12218:
12207:
12201:
12198:
12192:
12184:
12178:
12175:
12169:
12159:
12153:
12150:
12144:
12136:
12130:
12127:
12121:
12106:
12103:
12096:
12092:
12084:
12080:
12073:
12070:
12065:
12058:
12054:
12045:
12041:
12034:
12030:
12024:
12020:
12012:
12006:
12003:
11997:
11991:
11988:
11982:
11976:
11973:
11967:
11961:
11958:
11951:
11945:
11941:
11935:
11924:
11918:
11915:
11909:
11901:
11895:
11892:
11886:
11876:
11870:
11867:
11861:
11853:
11847:
11844:
11838:
11823:
11820:
11815:
11808:
11805:
11800:
11794:
11792:
11786:
11782:
11779:
11776:
11773:
11770:
11767:
11761:
11758:
11752:
11749:
11745:
11741:
11729:
11711:
11707:
11701:
11689:
11674:
11667:
11663:
11653:
11650:
11645:
11638:
11633:
11626:
11619:
11616:
11611:
11602:
11598:
11593:
11583:
11580:
11575:
11568:
11563:
11555:
11552:
11547:
11543:
11539:
11536:
11533:
11530:
11527:
11524:
11518:
11515:
11509:
11506:
11502:
11498:
11490:
11471:
11463:
11459:
11450:
11435:
11430:
11425:
11413:
11405:
11402:
11398:
11392:
11388:
11385:
11380:
11377:
11371:
11366:
11354:
11346:
11342:
11336:
11332:
11329:
11326:
11322:
11317:
11305:
11297:
11294:
11290:
11284:
11280:
11277:
11272:
11269:
11263:
11258:
11246:
11238:
11234:
11228:
11224:
11221:
11217:
11208:
11198:
11190:
11187:
11183:
11178:
11174:
11170:
11167:
11164:
11161:
11158:
11155:
11149:
11146:
11140:
11137:
11133:
11129:
11121:
11102:
11096:
11073:
11067:
11059:
11044:
11039:
11032:
11028:
11018:
11013:
11005:
11001:
10998:
10992:
10988:
10981:
10973:
10970:
10965:
10962:
10958:
10949:
10945:
10940:
10932:
10927:
10922:
10913:
10909:
10906:
10900:
10896:
10892:
10885:
10878:
10874:
10870:
10867:
10864:
10861:
10858:
10855:
10849:
10846:
10840:
10837:
10833:
10829:
10821:
10802:
10794:
10791:
10788:
10784:
10775:
10761:
10755:
10752:
10749:
10746:
10743:
10740:
10737:
10734:
10731:
10725:
10717:
10713:
10699:
10693:
10690:
10687:
10684:
10681:
10678:
10672:
10669:
10663:
10660:
10654:
10646:
10642:
10628:
10622:
10619:
10616:
10613:
10610:
10607:
10604:
10601:
10598:
10592:
10584:
10580:
10579:
10578:
10561:
10558:
10555:
10552:
10549:
10546:
10543:
10540:
10537:
10531:
10517:
10514:
10509:
10507:
10503:
10498:
10496:
10492:
10476:
10468:
10463:
10459:
10455:
10445:
10442:
10438:
10422:
10419:
10416:
10396:
10393:
10390:
10367:
10364:
10361:
10358:
10355:
10347:
10343:
10315:
10312:
10309:
10304:
10301:
10298:
10295:
10292:
10289:
10286:
10281:
10278:
10273:
10270:
10267:
10260:
10257:
10254:
10249:
10246:
10243:
10240:
10234:
10229:
10226:
10203:
10200:
10197:
10194:
10191:
10183:
10179:
10175:
10172:
10152:
10132:
10112:
10092:
10083:
10066:
10063:
10060:
10057:
10054:
10046:
10042:
10038:
10035:
10005:
10002:
9999:
9992:
9989:
9980:
9977:
9974:
9966:
9963:
9957:
9954:
9949:
9946:
9938:
9933:
9930:
9926:
9921:
9917:
9914:
9908:
9905:
9902:
9899:
9896:
9893:
9873:
9869:
9864:
9858:
9855:
9852:
9847:
9844:
9837:
9834:
9831:
9828:
9823:
9820:
9812:
9808:
9805:
9802:
9799:
9796:
9793:
9790:
9786:
9782:
9779:
9776:
9771:
9768:
9762:
9757:
9751:
9748:
9743:
9740:
9720:
9717:
9714:
9706:
9692:
9686:
9682:
9679:
9676:
9669:
9664:
9654:
9651:
9648:
9642:
9639:
9636:
9630:
9627:
9621:
9610:
9607:
9595:
9589:
9586:
9572:
9569:
9566:
9560:
9554:
9551:
9541:
9538:
9534:
9528:
9522:
9518:
9514:
9511:
9507:
9502:
9499:
9479:
9476:
9473:
9465:
9450:
9443:
9426:
9419:
9416:
9410:
9404:
9401:
9395:
9391:
9370:
9362:
9361:
9360:
9342:
9336:
9313:
9305:
9302:
9298:
9283:
9269:
9266:
9263:
9260:
9257:
9249:
9245:
9235:
9215:
9212:
9209:
9206:
9195:
9192:
9189:
9186:
9181:
9176:
9173:
9168:
9160:
9157:
9149:
9143:
9140:
9137:
9131:
9125:
9122:
9119:
9106:
9103:
9100:
9096:
9090:
9087:
9082:
9080:
9071:
9064:
9061:
9058:
9055:
9044:
9041:
9038:
9035:
9030:
9025:
9021:
9018:
9012:
9004:
9001:
8994:
8986:
8983:
8967:
8964:
8961:
8957:
8952:
8943:
8940:
8935:
8933:
8925:
8917:
8913:
8888:
8885:
8880:
8876:
8872:
8849:
8845:
8841:
8838:
8835:
8832:
8826:
8823:
8820:
8817:
8808:
8806:
8803: −
8802:
8798:
8795: −
8794:
8790:
8786:
8782:
8779: ≠
8778:
8761:
8754:
8751:
8748:
8745:
8734:
8731:
8728:
8725:
8720:
8714:
8711:
8708:
8704:
8699:
8691:
8688:
8681:
8673:
8670:
8654:
8651:
8648:
8644:
8639:
8630:
8627:
8622:
8616:
8613:
8610:
8607:
8604:
8601:
8598:
8595:
8592:
8586:
8560:
8557:
8554:
8551:
8543:
8539:
8535:
8532:
8511:
8507:
8504:
8497:
8494:
8487:
8477:
8473:
8469:
8466:
8450:
8447:
8444:
8440:
8431:
8428:
8425:
8419:
8416:
8412:
8401:
8397:
8392:
8383:
8380:
8375:
8369:
8366:
8363:
8360:
8357:
8354:
8351:
8348:
8345:
8339:
8331:
8330:Taylor series
8326:
8313:
8309:
8305:
8302:
8290:
8287:
8284:
8281:
8273:
8265:
8262:
8256:
8252:
8243:
8240:
8237:
8231:
8228:
8224:
8213:
8209:
8204:
8195:
8192:
8187:
8181:
8178:
8175:
8172:
8169:
8166:
8163:
8160:
8157:
8151:
8137:
8133:
8130: −
8129:
8125:
8121:
8117:
8109:
8105:
8101:
8085:
8082:
8079:
8059:
8055:
8051:
8048:
8045:
8037:
8034:
8030:
8014:
8011:
8008:
7988:
7985:
7982:
7974:
7973:
7972:
7964:
7962:
7960:
7959:
7955:
7952:
7950:
7949:
7945:
7942:
7939:
7938:
7935:
7934:
7928:
7927:
7923:
7920:
7918:
7917:
7913:
7910:
7907:
7904:
7901:
7887:
7879:
7875:
7872:
7869:
7866:
7863:
7860:
7857:
7855:
7853:
7852:
7837:
7828:
7826:
7825:
7822:
7820:
7816:
7812:
7808:
7792:
7783:
7780:
7772:
7768:
7764:
7748:
7745:
7742:
7722:
7718:
7714:
7711:
7708:
7700:
7697:
7693:
7689:
7673:
7670:
7667:
7647:
7644:
7641:
7633:
7618:
7610:
7606:
7602:
7598:
7582:
7579:
7576:
7568:
7567:
7566:
7564:
7560:
7556:
7552:
7530:
7527:
7524:
7518:
7510:
7505:
7486:
7483:
7480:
7457:
7454:
7451:
7445:
7437:
7433:
7428:
7422:Special cases
7419:
7417:
7413:
7406:
7403:
7399:
7395:
7391:
7386:
7382:
7379:
7376:
7375:
7370:
7365:
7361:
7356:
7352:
7345:
7340:
7336:
7333:
7332:
7331:
7330:
7327:
7322:
7318:
7313:
7309:
7306:
7305:and constants
7299:
7292:
7285:
7282:
7279:
7276:
7273:
7272:
7271:
7268:
7266:
7262:
7258:
7254:
7250:
7246:
7242:
7238:
7228:
7225:
7203:
7198:
7194:
7190:
7185:
7180:
7176:
7168:
7163:
7159:
7153:
7149:
7145:
7140:
7135:
7131:
7125:
7121:
7114:
7112:
7107:
7097:
7094:
7088:
7082:
7077:
7073:
7069:
7064:
7059:
7055:
7050:
7045:
7043:
7038:
7029:
7025:
7021:
7016:
7012:
7008:
7006:
7001:
6977:
6969:
6965:
6956:
6942:
6932:
6926:
6923:
6918:
6908:
6903:
6899:
6888:
6884:
6880:
6877:
6868:
6862:
6859:
6854:
6844:
6839:
6835:
6824:
6820:
6816:
6813:
6808:
6798:
6793:
6789:
6780:
6775:
6765:
6760:
6756:
6747:
6742:
6738:
6734:
6731:
6728:
6723:
6719:
6715:
6712:
6708:
6704:
6701:
6681:
6661:
6649:
6645:
6630:
6627:
6624:
6616:
6612:
6608:
6607:leptokurtotic
6604:
6601:
6597:
6596:
6590:
6573:
6565:
6561:
6537:
6534:
6531:
6525:
6502:
6494:
6476:
6462:
6459:
6456:
6448:
6428:
6425:
6422:
6414:
6407:
6397:
6389:
6385:
6373:
6370:
6358:
6351:
6348:
6343:
6340:
6337:
6332:
6327:
6322:
6319:
6314:
6308:
6304:
6298:
6295:
6282:
6279:
6267:
6261:
6256:
6248:
6239:
6228:
6224:
6218:
6215:
6209:
6203:
6200:
6188:
6184:
6176:
6175:
6174:
6172:
6168:
6152:
6148:
6144:
6141:
6138:
6115:
6112:
6109:
6106:
6098:
6091:
6087:
6083:
6079:
6075:
6072:
6069:
6063:
6055:
6051:
6045:
6042:
6039:
6035:
6024:
6020:
6012:
6011:
6010:
6007:
5991:
5987:
5978:
5962:
5959:
5956:
5936:
5933:
5928:
5924:
5900:
5897:
5892:
5888:
5884:
5881:
5872:
5869:
5851:
5835:
5830:
5808:
5805:
5802:
5797:
5793:
5769:
5766:
5763:
5759:
5754:
5750:
5746:
5743:
5735:
5727:
5724:
5717:
5713:
5709:
5706:
5700:
5696:
5690:
5686:
5682:
5672:
5668:
5664:
5661:
5650:
5646:
5642:
5638:
5632:
5627:
5623:
5618:
5612:
5609:
5604:
5600:
5596:
5593:
5584:
5581:
5564:
5563:
5562:
5548:
5545:
5537:
5534:is a shifted
5518:
5515:
5510:
5506:
5502:
5499:
5490:
5487:
5469:
5450:
5441:
5438:
5433:
5410:
5407:
5402:
5399:
5390:
5376:
5354:
5350:
5323:
5318:
5314:
5287:
5284:
5281:
5273:
5270:
5267:
5264:
5242:
5238:
5234:
5231:
5223:
5219:
5211:
5207:
5203:
5200:
5196:
5191:
5185:
5181:
5172:
5168:
5164:
5161:
5157:
5144:
5141:
5129:
5124:
5118:
5115:
5110:
5106:
5102:
5099:
5091:
5075:
5074:
5073:
5071:
5052:
5049:
5046:
5038:
5035:
5032:
5024:
5020:
5015:
5012:
5007:
5001:
4997:
4991:
4988:
4979:
4974:
4971:
4966:
4958:
4953:
4947:
4939:
4923:
4922:
4921:
4919:
4918:
4901:
4898:
4895:
4873:
4869:
4865:
4861:
4857:
4854:
4833:
4828:
4825:
4820:
4814:
4810:
4804:
4801:
4791:
4772:
4764:
4760:
4756:
4751:
4747:
4724:
4720:
4714:
4709:
4706:
4703:
4699:
4695:
4692:
4683:
4669:
4660:
4643:
4640:
4637:
4628:
4624:
4616:
4610:
4607:
4601:
4598:
4592:
4589:
4583:
4580:
4573:
4569:
4561:
4550:
4546:
4535:
4531:
4525:
4522:
4517:
4515:
4502:
4499:
4496:
4487:
4483:
4475:
4469:
4466:
4463:
4457:
4454:
4448:
4445:
4439:
4436:
4429:
4425:
4417:
4411:
4408:
4405:
4401:
4390:
4386:
4380:
4377:
4372:
4370:
4361:
4357:
4354:
4346:
4336:
4328:
4325:
4321:
4315:
4312:
4309:
4305:
4291:
4287:
4282:
4273:
4270:
4265:
4263:
4255:
4247:
4243:
4218:
4215:
4209:
4203:
4200:
4179:
4173:
4163:
4155:
4152:
4148:
4144:
4141:
4138:
4132:
4129:
4126:
4120:
4097:
4093:
4089:
4086:
4083:
4080:
4074:
4071:
4068:
4065:
4042:
4039:
4036:
4015:
4011:
4008:
4003:
3999:
3995:
3990:
3985:
3981:
3978:
3972:
3967:
3964:
3961:
3958:
3955:
3952:
3949:
3946:
3942:
3938:
3935:
3929:
3921:
3917:
3909:
3908:
3907:
3905:
3901:
3897:
3881:
3878:
3875:
3855:
3852:
3849:
3835:
3825:
3821:
3817:
3801:
3798:
3795:
3786:
3772:
3769:
3766:
3758:
3754:
3738:
3735:
3732:
3729:
3709:
3706:
3703:
3695:
3678:
3672:
3665:
3662:
3659:
3646:
3641:
3637:
3634:
3628:
3624:
3621:
3615:
3609:
3603:
3600:
3597:
3594:
3586:
3582:
3577:
3568:
3565:
3562:
3552:
3543:
3538:
3532:
3526:
3506:
3503:
3500:
3491:
3475:
3472:
3466:
3460:
3448:
3444:
3433:
3428:
3425:
3422:
3418:
3413:
3408:
3405:
3382:
3379:
3376:
3373:
3370:
3367:
3364:
3361:
3358:
3354:
3350:
3344:
3338:
3335:
3324:
3305:
3302:
3297:
3293:
3284:
3280:
3274:
3269:
3266:
3263:
3259:
3255:
3252:
3244:
3240:
3236:
3217:
3214:
3211:
3208:
3205:
3202:
3199:
3196:
3193:
3187:
3173:
3160:
3156:
3150:
3146:
3143:
3136:
3132:
3129:
3126:
3123:
3120:
3117:
3109:
3093:
3090:
3087:
3067:
3045:
3041:
3038:
3035:
3029:
3026:
3017:
2997:
2994:
2991:
2986:
2983:
2980:
2975:
2972:
2967:
2964:
2959:
2955:
2952:
2949:
2939:
2936:
2933:
2926:
2922:
2919:
2916:
2907:
2902:
2899:
2876:
2873:
2870:
2867:
2864:
2861:
2858:
2855:
2852:
2846:
2837:
2835:
2831:
2827:
2823:
2807:
2787:
2778:
2758:
2755:
2752:
2742:
2739:
2731:
2728:
2723:
2720:
2715:
2708:
2705:
2702:
2696:
2690:
2686:
2683:
2676:
2672:
2669:
2665:
2661:
2658:
2653:
2650:
2647:
2637:
2634:
2625:
2618:
2613:
2589:
2584:
2574:
2568:
2565:
2562:
2559:
2556:
2553:
2549:
2543:
2533:
2530:
2522:
2519:
2516:
2513:
2509:
2505:
2502:
2499:
2493:
2490:
2487:
2484:
2481:
2478:
2475:
2472:
2469:
2463:
2443:
2440:
2437:
2428:
2419:
2405:
2397:
2392:
2390:
2374:
2371:
2368:
2348:
2345:
2342:
2333:
2331:
2327:
2323:
2319:
2303:
2300:
2297:
2288:
2286:
2282:
2277:
2272:
2270:
2266:
2250:
2230:
2220:
2218:
2214:
2198:
2195:
2192:
2184:
2180:
2177:, called the
2161:
2158:
2155:
2152:
2146:
2143:
2135:
2131:
2109:
2106:
2103:
2093:
2085:
2082:
2077:
2074:
2069:
2062:
2059:
2056:
2050:
2045:
2041:
2038:
2032:
2028:
2025:
2019:
2014:
1999:
1993:
1974:
1969:
1959:
1953:
1950:
1947:
1944:
1941:
1938:
1934:
1928:
1918:
1915:
1907:
1904:
1901:
1898:
1894:
1890:
1887:
1884:
1878:
1875:
1872:
1869:
1866:
1863:
1860:
1857:
1854:
1848:
1840:
1835:
1822:
1819:
1816:
1810:
1807:
1804:
1800:
1793:
1787:
1774:
1770:
1766:
1760:
1754:
1731:
1725:
1717:
1698:
1692:
1685:
1680:
1666:
1646:
1638:
1634:
1630:
1625:
1623:
1619:
1615:
1606:
1601:
1596:
1589:
1583:
1579:
1570:
1563:
1556:
1549:
1544:
1539:
1533:
1529:
1521:
1512:
1506:
1504:
1494:
1480:
1477:
1474:
1471:
1468:
1460:
1456:
1451:
1447:
1443:
1427:
1424:
1421:
1413:
1397:
1394:
1391:
1383:
1379:
1363:
1360:
1357:
1349:
1333:
1330:
1327:
1324:
1321:
1301:
1292:
1290:
1286:
1282:
1278:
1274:
1271:
1267:
1264:
1260:
1256:
1252:
1248:
1220:
1217:
1214:
1199:
1191:
1188:
1182:
1179:
1173:
1166:
1163:
1160:
1147:
1143:
1140:
1133:
1130:
1124:
1119:
1108:
1094:
1074:
1068:
1065:
1062:
1059:
1056:
1053:
1044:
1034:
1030:
1022:
1019:
1016:
1013:
1001:
991:
987:
969:
966:
963:
960:
957:
954:
951:
948:
945:
942:
939:
936:
909:
906:
903:
900:
895:
892:
888:
884:
881:
878:
875:
872:
861:
840:
837:
834:
831:
828:
825:
822:
819:
816:
813:
810:
807:
777:
773:
769:
766:
760:
757:
752:
748:
742:
738:
734:
731:
728:
717:
696:
693:
690:
661:
657:
651:
647:
643:
640:
637:
626:
617:
613:
607:
603:
587:
584:
581:
571:
567:
551:
548:
545:
535:
531:
515:
512:
509:
501:
495:
491:
475:
472:
469:
461:
456:
452:
436:
433:
430:
422:
417:
413:
397:
394:
391:
383:
378:
374:
368:
364:
358:
354:
350:
346:
343:
330:
327:
324:
321:
301:
298:
295:
287:
283:
279:
266:
263:
260:
240:
237:
234:
226:
222:
216:
212:
209:
205:
202:
200:
196:
192:is undefined)
191:
175:
151:
148:
145:
139:
136:
126:
122:
114:
95:
84:
76:
68:
49:
38:
30:
22:
16462:
16450:
16416:Multivariate
16415:
16403:
16391:
16386:Wrapped Lévy
16346:
16294:Matrix gamma
16287:
16267:
16255:Normal-gamma
16248:
16214:Continuous:
16213:
16184:
16129:Tukey lambda
16116:
16108:
16103:-exponential
16100:
16092:
16083:
16074:
16065:
16059:-exponential
16056:
16000:
15993:
15967:
15934:
15896:
15883:
15810:Poly-Weibull
15755:Log-logistic
15715:
15714:Hotelling's
15646:
15488:Logit-normal
15362:Gauss–Kuzmin
15357:Flory–Schulz
15238:with finite
15152:
15148:
15138:
15129:
15125:
15119:
15110:
15104:
15071:
15067:
15033:
15029:
15009:
14989:
14982:
14957:
14953:
14947:
14914:
14910:
14904:
14884:
14877:
14842:
14838:
14832:
14801:
14794:
14773:
14765:
14740:
14736:
14730:
14713:
14709:
14696:
14677:
14673:
14663:
14630:
14626:
14592:
14588:
14582:
14572:
14542:
14535:
14526:
14520:
14511:
14505:
14491:(1): 78–91.
14488:
14484:
14474:
14465:
14395:
14391:
14357:
14353:
14308:
14269:
14247:. Retrieved
14243:
14233:
14222:. Retrieved
14215:the original
14165:
14159:
14125:
14121:
14115:
14090:
14086:
14080:
14055:
14052:Econometrica
14051:
14045:
14028:
14022:
13997:
13993:
13945:'stabledist'
13924:
13857:Lévy process
10718:is given by
10647:is given by
10585:is given by
10523:
10510:
10502:spectroscopy
10499:
10451:
10448:Applications
10084:
10027:
9463:with mean 1;
9329:nor the CDF
9289:
9241:
8809:
8804:
8800:
8796:
8792:
8784:
8780:
8776:
8327:
8143:
8131:
8127:
8123:
8115:
8113:
8107:
8103:
8032:
7970:
7956:
7946:
7931:
7924:
7914:
7818:
7814:
7784:
7776:
7770:
7766:
7695:
7691:
7608:
7604:
7600:
7554:
7550:
7548:
7508:
7435:
7431:
7415:
7411:
7409:
7404:
7401:
7397:
7393:
7389:
7384:
7380:
7377:
7368:
7363:
7359:
7354:
7350:
7343:
7338:
7334:
7325:
7320:
7316:
7311:
7307:
7304:
7297:
7290:
7283:
7280:
7277:
7274:
7269:
7234:
7226:
6967:
6963:
6957:
6653:
6477:
6443:
6130:
6008:
5785:Its mean is
5784:
5391:
5302:
5067:
4915:
4789:
4684:
4661:
4057:
3903:
3899:
3895:
3841:
3823:
3819:
3815:
3787:
3752:
3492:
3322:
3242:
3238:
3234:
3179:
3107:
3018:
2838:
2833:
2829:
2825:
2821:
2779:
2429:
2425:
2395:
2393:
2388:
2334:
2329:
2325:
2321:
2289:
2284:
2280:
2273:
2268:
2264:
2221:
2178:
2133:
2129:
1996:is just the
1991:
1838:
1836:
1681:
1636:
1632:
1626:
1611:
1604:
1599:
1594:
1587:
1581:
1577:
1568:
1561:
1554:
1547:
1542:
1537:
1531:
1519:
1510:
1500:
1293:
1284:
1280:
1254:
1251:distribution
1244:
994:
499:
459:
420:
381:
348:
344:
285:
281:
280:
224:
220:
219:
206:∈ (−∞, ∞) —
203:
194:
129:
16500:Exponential
16349:directional
16338:Directional
16225:Generalized
16196:Multinomial
16151:continuous-
16091:Kaniadakis
16082:Kaniadakis
16073:Kaniadakis
16064:Kaniadakis
16055:Kaniadakis
16007:Tracy–Widom
15984:Skew normal
15966:Noncentral
15750:Log-Laplace
15728:Generalized
15709:Half-normal
15675:Generalized
15639:Logarithmic
15624:Exponential
15578:Chi-squared
15518:U-quadratic
15483:Kumaraswamy
15425:Continuous
15372:Logarithmic
15267:Categorical
13852:Lévy flight
11726:denote the
11118:denote the
6648:convolution
1263:independent
197:∈ (0, ∞) —
16594:Power laws
16578:Categories
16495:Elliptical
16451:Degenerate
16437:Degenerate
16185:Discrete:
16144:univariate
15999:Student's
15954:Asymmetric
15933:Johnson's
15861:supported
15805:Phase-type
15760:Log-normal
15745:Log-Cauchy
15735:Kolmogorov
15653:Noncentral
15583:Noncentral
15563:Beta prime
15513:Triangular
15508:Reciprocal
15478:Irwin–Hall
15427:univariate
15407:Yule–Simon
15289:Rademacher
15231:univariate
15132:: 163–167.
14954:Statistics
14249:2018-10-18
14224:2009-02-21
13977:References
7265:Kolmogorov
7253:Kolmogorov
6593:Properties
3757:heavy tail
1620:, and the
1612:Since the
1497:Definition
1455:Mandelbrot
1442:attractors
125:Parameters
42:Symmetric
16220:Dirichlet
16201:Dirichlet
16111:-Gaussian
16086:-Logistic
15923:Holtsmark
15895:Gaussian
15882:Fisher's
15865:real line
15367:Geometric
15347:Delaporte
15252:Bernoulli
15229:Discrete
14993:. Wiley.
14974:0233-1888
14847:CiteSeerX
14757:0162-1459
14655:0001-8732
14609:0040-585X
14405:1007.0193
14374:0882-0287
14336:226648987
13931:libstable
13816:≥
13753:−
13715:
13691:π
13629:−
13561:
13537:π
13446:−
13381:−
13361:−
13353:
13329:π
13246:−
13196:−
13178:−
13153:
13129:π
13021:μ
12936:−
12825:π
12789:Γ
12724:−
12598:π
12579:−
12533:−
12429:π
12405:Γ
12189:Γ
12166:Γ
12141:Γ
12118:Γ
12107:π
12059:−
11906:Γ
11883:Γ
11858:Γ
11835:Γ
11824:π
11556:π
11406:π
11386:−
11333:
11298:π
11278:−
11225:
11191:π
11002:π
10933:π
10910:π
10901:−
10882:ℜ
10795:ν
10789:μ
10562:μ
10550:β
10544:α
10516:effects.
10477:α
10417:β
10391:α
10368:μ
10356:β
10348:α
10310:α
10305:μ
10296:
10287:β
10282:π
10258:≠
10255:α
10250:μ
10192:β
10184:α
10176:∼
10153:μ
10133:β
10093:α
10055:β
10047:α
10039:∼
10000:α
9990:π
9978:≠
9975:α
9967:ζ
9964:−
9958:
9950:α
9931:ξ
9918:α
9915:π
9909:
9903:β
9900:−
9894:ζ
9856:β
9845:π
9835:
9821:π
9809:
9803:β
9800:−
9794:
9780:β
9769:π
9752:ξ
9733:compute:
9715:α
9687:α
9683:α
9680:−
9655:ξ
9643:α
9640:−
9631:
9611:α
9590:
9573:ξ
9561:α
9555:
9542:α
9519:ζ
9492:compute:
9477:≠
9474:α
9417:π
9402:π
9396:−
9303:−
9267:≤
9264:α
9207:α
9201:Γ
9187:α
9150:π
9138:α
9126:
9120:−
9112:∞
9097:∑
9091:π
9056:α
9050:Γ
9036:α
9019:−
8984:−
8973:∞
8958:∑
8949:ℜ
8944:π
8918:α
8881:α
8842:π
8839:α
8833:−
8827:
8746:α
8740:Γ
8726:α
8715:μ
8712:−
8671:−
8660:∞
8645:∑
8636:ℜ
8631:π
8617:μ
8605:β
8599:α
8564:Φ
8561:β
8555:−
8544:α
8478:α
8467:−
8456:∞
8441:∑
8432:μ
8429:−
8407:∞
8398:∫
8389:ℜ
8384:π
8370:μ
8358:β
8352:α
8294:Φ
8291:β
8285:−
8274:α
8257:−
8244:μ
8241:−
8219:∞
8210:∫
8201:ℜ
8196:π
8182:μ
8170:β
8164:α
8080:β
8046:α
8009:β
7983:α
7888:β
7838:α
7793:α
7743:β
7709:α
7668:β
7642:α
7619:β
7607:and mean
7577:α
7525:α
7519:−
7481:α
7452:α
7446:−
7257:Paul Lévy
7241:Lindeberg
7204:α
7186:α
7169:α
7150:β
7141:α
7122:β
7108:β
7098:α
7083:α
7065:α
7026:μ
7013:μ
7002:μ
6978:β
6939:Φ
6927:
6919:α
6885:β
6875:Φ
6863:
6855:α
6821:β
6809:α
6781:−
6776:α
6748:−
6739:μ
6720:μ
6705:
6682:α
6662:α
6625:α
6566:α
6526:−
6503:ν
6495:α
6457:α
6423:α
6408:α
6390:−
6374:α
6363:Γ
6359:α
6341:ν
6323:ν
6309:α
6299:ν
6283:α
6272:Γ
6268:α
6249:ν
6229:−
6204:ν
6194:∞
6185:∫
6153:ν
6110:α
6092:α
6084:−
6056:α
6040:−
6030:∞
6021:∫
5988:ν
5957:θ
5925:ν
5901:θ
5889:ν
5882:ν
5836:θ
5809:θ
5794:ν
5764:θ
5751:ν
5744:ν
5728:θ
5714:ν
5710:−
5707:ν
5701:−
5669:ν
5665:−
5662:ν
5639:θ
5633:π
5613:θ
5601:ν
5594:ν
5549:θ
5519:θ
5507:ν
5500:ν
5400:α
5377:θ
5351:ν
5327:∞
5315:ν
5282:α
5265:θ
5239:ν
5232:ν
5208:ν
5204:−
5201:ν
5197:θ
5186:α
5169:ν
5165:−
5162:ν
5145:α
5134:Γ
5130:α
5119:θ
5107:ν
5100:ν
5092:α
5047:α
5033:ν
5016:ν
5002:α
4992:ν
4975:α
4963:Γ
4959:α
4948:ν
4940:α
4874:α
4855:ν
4829:ν
4815:α
4805:ν
4765:α
4757:∼
4700:∑
4629:α
4611:
4602:
4584:
4574:α
4551:−
4541:∞
4532:∫
4526:π
4488:α
4470:
4464:−
4458:
4440:
4430:α
4412:
4406:−
4396:∞
4387:∫
4381:π
4347:α
4326:−
4300:∞
4295:∞
4292:−
4288:∫
4279:ℜ
4274:π
4248:α
4204:
4174:α
4153:−
4145:
4133:α
4121:φ
4090:π
4087:α
4081:−
4075:
4037:α
4004:α
3982:π
3979:α
3968:
3953:α
3922:α
3876:β
3850:α
3796:α
3767:α
3736:±
3730:β
3707:≥
3704:α
3673:π
3660:α
3654:Γ
3638:α
3635:π
3625:
3616:β
3604:
3587:α
3569:α
3539:∼
3501:α
3476:α
3461:α
3419:∑
3371:β
3365:α
3306:μ
3303:−
3260:∑
3218:μ
3206:β
3200:α
3147:α
3144:π
3133:
3127:γ
3124:β
3121:−
3118:δ
3088:α
3068:α
3046:γ
3042:δ
3039:−
2992:α
2987:γ
2984:
2976:π
2968:β
2965:−
2960:γ
2956:μ
2953:−
2937:≠
2934:α
2927:γ
2923:μ
2920:−
2865:β
2859:α
2808:β
2788:α
2753:α
2740:γ
2732:
2724:π
2716:−
2706:≠
2703:α
2687:α
2684:π
2673:
2659:−
2654:α
2651:−
2635:γ
2611:Φ
2581:Φ
2569:
2563:β
2557:−
2544:α
2531:γ
2523:−
2520:δ
2506:
2494:δ
2488:γ
2482:β
2476:α
2464:φ
2438:α
2406:α
2369:β
2343:α
2298:β
2251:β
2231:α
2193:α
2153:−
2147:∈
2144:β
2104:α
2086:
2078:π
2070:−
2060:≠
2057:α
2042:α
2039:π
2029:
2012:Φ
1966:Φ
1954:
1948:β
1942:−
1929:α
1908:−
1905:μ
1891:
1879:μ
1867:β
1861:α
1849:φ
1783:∞
1778:∞
1775:−
1771:∫
1755:φ
1693:φ
1667:α
1647:β
1475:α
1425:≤
1422:α
1392:α
1358:α
1331:≤
1328:α
1302:α
1289:Paul Lévy
1215:α
1192:
1183:π
1174:−
1164:≠
1161:α
1144:α
1141:π
1134:
1117:Φ
1081:Φ
1069:
1063:β
1057:−
1045:α
1023:−
1020:μ
955:−
949:β
937:α
907:
893:−
889:π
879:−
876:μ
826:−
820:β
811:≠
808:α
770:α
767:π
761:
753:α
743:α
735:−
732:μ
691:α
641:μ
582:α
546:α
510:α
470:β
431:β
392:α
351:otherwise
328:−
322:β
296:α
261:β
235:α
227:, +∞) if
176:β
140:∈
137:α
96:α
50:α
16553:Category
16485:Circular
16478:Families
16463:Singular
16442:singular
16206:Negative
16153:discrete
16119:-Weibull
16077:-Weibull
15961:Logistic
15845:Discrete
15815:Rayleigh
15795:Nakagami
15718:-squared
15692:Gompertz
15541:interval
15277:Negative
15262:Binomial
15177:18754585
15096:11970402
14438:27497684
14430:21231282
14182:53004476
13961:package.
13943:Package
13846:See also
13048:, then:
11730:, then:
11122:, then:
10822:, then:
10577:, then:
10441:Gaussian
7349:+ ... +
7315:> 0,
7261:Gnedenko
7237:Berstein
4029:, where
2183:skewness
1382:variance
1287:, after
1273:location
1211:if
1157:if
534:Skewness
494:Variance
190:skewness
16563:Commons
16535:Wrapped
16530:Tweedie
16525:Pearson
16520:Mixture
16427:Bingham
16326:Complex
16316:Inverse
16306:Wishart
16299:Inverse
16286:Matrix
16260:Inverse
16176:(joint)
16095:-Erlang
15949:Laplace
15840:Weibull
15697:Shifted
15680:Inverse
15665:Fréchet
15588:Inverse
15523:Uniform
15443:Arcsine
15402:Skellam
15397:Poisson
15320:support
15294:Soliton
15247:Benford
15240:support
15157:Bibcode
15076:Bibcode
15038:Bibcode
14939:9961762
14919:Bibcode
14869:9500064
14680:: 908.
14635:Bibcode
14497:2245503
14410:Bibcode
14142:2350971
14107:2350970
14072:1911802
14037:1417531
14014:2525289
13957:in the
13925:randist
11487:be the
10458:moments
7779:mixture
7414:, then
6518:is the
5466:is the
4788:, then
3827:√
2603:where:
2213:moments
1535:. Then
1376:to the
1261:of two
606:Entropy
574:0 when
538:0 when
284:∈ (-∞,
215:Support
16469:Cantor
16311:Normal
16142:Mixed
16068:-Gamma
15994:Stable
15944:Landau
15918:Gumbel
15872:Cauchy
15800:Pareto
15612:Erlang
15593:Scaled
15548:Benini
15387:Panjer
15175:
15094:
14997:
14972:
14937:
14892:
14867:
14849:
14820:
14782:
14755:
14653:
14607:
14550:
14495:
14436:
14428:
14372:
14334:
14324:
14284:
14180:
14140:
14105:
14070:
14035:
14012:
13972:Nolan.
13951:Python
10383:. For
9955:arctan
9886:where
8525:where
7303:, ...
7249:Feller
6958:Since
4888:. Set
4847:where
4739:where
3519:, by:
2832:(like
2824:(like
2391:, ∞).
1988:where
1616:, the
1590:> 0
1557:> 0
1550:> 0
1543:stable
1501:A non-
1350:, and
1281:stable
1255:stable
1109:where
416:Median
27:Stable
16191:Ewens
16017:Voigt
15989:Slash
15770:Lomax
15765:Log-t
15670:Gamma
15617:Hyper
15607:Davis
15602:Dagum
15458:Bates
15448:ARGUS
15332:Borel
15173:S2CID
14865:S2CID
14806:(PDF)
14706:(PDF)
14493:JSTOR
14434:S2CID
14400:arXiv
14332:S2CID
14218:(PDF)
14211:(PDF)
14178:S2CID
14138:JSTOR
14103:JSTOR
14068:JSTOR
14010:JSTOR
13965:Julia
13959:SciPy
13933:is a
13044:be a
10818:be a
10409:(and
10219:then
5916:with
5392:When
3842:When
3788:When
3398:with
3321:then
2335:When
1277:scale
1270:up to
1257:if a
929:when
800:when
683:when
502:when
462:when
423:when
384:when
288:] if
16440:and
16398:Kent
15825:Rice
15740:Lévy
15568:Burr
15498:PERT
15463:Beta
15412:Zeta
15304:Zipf
15221:list
15092:PMID
14995:ISBN
14970:ISSN
14935:PMID
14890:ISBN
14818:ISBN
14780:ISBN
14753:ISSN
14651:ISSN
14605:ISSN
14548:ISBN
14466:SSRN
14426:PMID
14370:ISSN
14322:ISBN
14282:ISBN
14033:OCLC
13915:The
13662:<
13002:Let
11690:Let
11451:Let
11089:and
11060:Let
10776:Let
10714:The
10643:The
10581:The
10513:CGRO
10145:and
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9466:for
9261:<
8865:and
8072:and
8038:For
8001:and
7975:For
7735:and
7701:For
7660:and
7634:For
7569:For
7392:) →
7358:) −
7326:with
7324:∈ ℝ
7263:and
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6609:and
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2800:and
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2267:and
2243:and
2196:<
2004:and
1998:sign
1990:sgn(
1682:The
1659:and
1635:and
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1552:and
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314:and
299:<
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15165:doi
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15111:VSP
15084:doi
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14962:doi
14927:doi
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14810:doi
14745:doi
14718:doi
14682:doi
14643:doi
14597:doi
14418:doi
14396:105
14362:doi
14314:doi
14274:doi
14170:doi
14130:doi
14095:doi
14060:doi
14002:doi
13712:exp
13558:exp
13350:exp
13150:exp
11330:cos
11222:sin
10335:is
10293:log
9906:tan
9832:cos
9806:log
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9628:cos
9587:cos
9552:sin
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9123:sin
8824:exp
8791:in
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7870:4/3
7864:2/3
7861:1/2
7858:1/3
7603:= 2
6970:or
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6860:sgn
6702:exp
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4437:sin
4142:exp
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2729:log
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2566:sgn
2503:exp
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2083:log
2026:tan
2000:of
1951:sgn
1888:exp
1607:= 0
1446:iid
1245:In
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1002:exp
862:exp
758:sec
718:exp
627:exp
616:MGF
367:CDF
357:PDF
223:∈ [
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7916:E
7911:s
7908:s
7905:s
7902:0
7867:1
7819:s
7815:E
7773:.
7771:μ
7767:c
7749:1
7746:=
7723:2
7719:/
7715:1
7712:=
7698:.
7696:μ
7692:c
7674:0
7671:=
7648:1
7645:=
7609:μ
7605:c
7601:σ
7583:2
7580:=
7555:x
7553:(
7551:f
7534:)
7531:1
7528:+
7522:(
7509:x
7487:2
7484:=
7461:)
7458:1
7455:+
7449:(
7436:x
7432:x
7416:Z
7412:Z
7398:y
7396:(
7394:F
7390:y
7388:(
7385:n
7381:F
7364:n
7360:b
7355:n
7351:X
7347:1
7344:X
7342:(
7339:n
7335:a
7321:n
7317:b
7312:n
7308:a
7301:3
7298:X
7294:2
7291:X
7287:1
7284:X
7278:Z
7199:2
7195:c
7191:+
7181:1
7177:c
7164:2
7160:c
7154:2
7146:+
7136:1
7132:c
7126:1
7115:=
7095:1
7089:)
7078:2
7074:c
7070:+
7060:1
7056:c
7051:(
7046:=
7039:c
7030:2
7022:+
7017:1
7009:=
6968:c
6964:μ
6960:Φ
6943:)
6936:)
6933:t
6930:(
6914:|
6909:t
6904:2
6900:c
6895:|
6889:2
6881:i
6878:+
6872:)
6869:t
6866:(
6850:|
6845:t
6840:1
6836:c
6831:|
6825:1
6817:i
6814:+
6804:|
6799:t
6794:2
6790:c
6785:|
6771:|
6766:t
6761:1
6757:c
6752:|
6743:2
6735:t
6732:i
6729:+
6724:1
6716:t
6713:i
6709:(
6650:.
6631:2
6628:=
6617:(
6602:.
6577:)
6574:x
6571:(
6562:L
6541:)
6538:1
6535:+
6532:n
6529:(
6506:)
6500:(
6489:N
6463:1
6445:"
6415:,
6403:|
6398:z
6394:|
6386:e
6379:)
6371:1
6366:(
6352:2
6349:1
6344:=
6338:d
6333:)
6328:)
6320:1
6315:(
6305:L
6296:1
6288:)
6280:1
6275:(
6262:(
6257:)
6244:|
6240:z
6236:|
6225:e
6219:2
6216:1
6210:(
6201:1
6189:0
6149:/
6145:1
6142:=
6139:x
6099:,
6088:z
6080:e
6076:=
6073:x
6070:d
6067:)
6064:x
6061:(
6052:L
6046:x
6043:z
6036:e
6025:0
5992:0
5960:=
5934:=
5929:0
5904:)
5898:,
5893:0
5885:;
5879:(
5873:2
5870:1
5863:N
5806:6
5803:+
5798:0
5760:,
5755:0
5736:,
5725:4
5718:0
5697:e
5691:2
5687:/
5683:1
5679:)
5673:0
5659:(
5651:2
5647:/
5643:3
5628:4
5624:1
5619:=
5616:)
5610:,
5605:0
5597:;
5591:(
5585:2
5582:1
5575:N
5546:4
5522:)
5516:,
5511:0
5503:;
5497:(
5491:2
5488:1
5481:N
5454:)
5451:x
5448:(
5442:2
5439:1
5434:L
5411:2
5408:1
5403:=
5355:0
5330:)
5324:,
5319:0
5311:[
5274:,
5271:0
5243:0
5224:,
5220:)
5212:0
5192:(
5182:L
5173:0
5158:1
5150:)
5142:1
5137:(
5125:=
5122:)
5116:,
5111:0
5103:;
5097:(
5086:N
5039:0
5025:,
5021:)
5013:1
5008:(
4998:L
4989:1
4980:)
4972:1
4967:(
4954:=
4951:)
4945:(
4934:N
4902:1
4899:=
4896:x
4870:/
4866:1
4862:N
4858:=
4834:)
4826:x
4821:(
4811:L
4802:1
4790:Y
4776:)
4773:x
4770:(
4761:L
4752:i
4748:X
4725:i
4721:X
4715:N
4710:1
4707:=
4704:i
4696:=
4693:Y
4670:x
4644:.
4641:t
4638:d
4634:)
4625:t
4620:)
4617:q
4614:(
4605:(
4596:)
4593:x
4590:t
4587:(
4570:t
4565:)
4562:q
4559:(
4547:e
4536:0
4523:2
4518:=
4503:,
4500:t
4497:d
4493:)
4484:t
4479:)
4476:q
4473:(
4461:(
4452:)
4449:x
4446:t
4443:(
4426:t
4421:)
4418:q
4415:(
4402:e
4391:0
4378:2
4373:=
4362:]
4358:t
4355:d
4342:|
4337:t
4333:|
4329:q
4322:e
4316:x
4313:t
4310:i
4306:e
4283:[
4271:1
4266:=
4259:)
4256:x
4253:(
4244:L
4219:0
4213:)
4210:q
4207:(
4180:)
4169:|
4164:t
4160:|
4156:q
4149:(
4139:=
4136:)
4130:;
4127:t
4124:(
4101:)
4098:2
4094:/
4084:i
4078:(
4069:=
4066:q
4016:)
4012:0
4009:,
4000:/
3996:1
3991:)
3986:2
3973:(
3962:,
3959:1
3956:,
3950:;
3947:x
3943:(
3939:f
3936:=
3933:)
3930:x
3927:(
3918:L
3904:μ
3896:μ
3882:1
3879:=
3856:1
3830:π
3824:c
3820:c
3816:x
3802:2
3799:=
3773:2
3753:μ
3739:1
3733:=
3710:1
3679:)
3669:)
3666:1
3663:+
3657:(
3647:)
3642:2
3629:(
3619:)
3613:)
3610:x
3607:(
3598:+
3595:1
3592:(
3583:c
3578:(
3566:+
3563:1
3558:|
3553:x
3549:|
3544:1
3536:)
3533:x
3530:(
3527:f
3507:2
3473:1
3467:)
3456:|
3449:i
3445:k
3440:|
3434:N
3429:1
3426:=
3423:i
3414:(
3409:=
3406:s
3386:)
3383:0
3380:,
3377:c
3374:,
3368:,
3362:;
3359:s
3355:/
3351:y
3348:(
3345:f
3339:s
3336:1
3323:Y
3309:)
3298:i
3294:X
3290:(
3285:i
3281:k
3275:N
3270:1
3267:=
3264:i
3256:=
3253:Y
3243:X
3239:Y
3235:X
3221:)
3215:,
3212:c
3209:,
3203:,
3197:;
3194:x
3191:(
3188:f
3161:.
3157:)
3151:2
3137:(
3108:μ
3094:1
3036:x
3030:=
3027:y
2998:1
2995:=
2973:2
2950:x
2940:1
2917:x
2908:{
2903:=
2900:y
2880:)
2877:0
2874:,
2871:1
2868:,
2862:,
2856:;
2853:y
2850:(
2847:f
2834:μ
2830:δ
2826:c
2822:γ
2759:1
2756:=
2747:|
2743:t
2736:|
2721:2
2709:1
2697:)
2691:2
2677:(
2666:)
2662:1
2648:1
2643:|
2638:t
2631:|
2626:(
2619:{
2614:=
2590:)
2585:)
2578:)
2575:t
2572:(
2560:i
2554:1
2550:(
2539:|
2534:t
2527:|
2517:t
2514:i
2510:(
2500:=
2497:)
2491:,
2485:,
2479:,
2473:;
2470:t
2467:(
2444:1
2441:=
2396:c
2389:μ
2375:1
2372:=
2349:1
2322:μ
2304:0
2301:=
2285:t
2281:t
2269:c
2265:μ
2199:2
2165:]
2162:1
2159:,
2156:1
2150:[
2134:R
2130:μ
2110:1
2107:=
2098:|
2094:t
2090:|
2075:2
2063:1
2051:)
2046:2
2033:(
2020:{
2015:=
2002:t
1994:)
1992:t
1975:)
1970:)
1963:)
1960:t
1957:(
1945:i
1939:1
1935:(
1924:|
1919:t
1916:c
1912:|
1902:t
1899:i
1895:(
1885:=
1882:)
1876:,
1873:c
1870:,
1864:,
1858:;
1855:t
1852:(
1839:X
1823:.
1820:x
1817:d
1811:t
1808:x
1805:i
1801:e
1797:)
1794:x
1791:(
1788:f
1767:=
1764:)
1761:t
1758:(
1735:)
1732:x
1729:(
1726:f
1702:)
1699:t
1696:(
1637:c
1633:μ
1609:.
1605:d
1595:d
1588:c
1582:d
1572:2
1565:1
1555:b
1548:a
1538:X
1532:X
1523:2
1520:X
1514:1
1511:X
1481:2
1469:1
1428:1
1398:2
1364:1
1361:=
1334:2
1322:0
1221:1
1218:=
1204:|
1200:t
1196:|
1180:2
1167:1
1148:2
1125:{
1120:=
1095:,
1090:]
1084:)
1078:)
1075:t
1072:(
1060:i
1054:1
1051:(
1040:|
1035:t
1031:c
1027:|
1017:t
1014:i
1008:[
982:,
970:0
964:t
961:,
958:1
952:=
946:,
943:1
940:=
915:)
910:t
901:t
896:1
885:2
882:c
873:t
868:(
853:,
841:0
835:t
832:,
829:1
823:=
817:,
814:1
786:)
781:)
778:2
774:/
764:(
749:t
739:c
729:t
724:(
709:,
697:2
694:=
669:)
662:2
658:t
652:2
648:c
644:+
638:t
633:(
588:2
585:=
552:2
549:=
516:2
513:=
500:c
498:2
476:0
473:=
460:μ
437:0
434:=
421:μ
398:1
382:μ
349:R
345:x
331:1
325:=
302:1
286:μ
282:x
267:1
264:=
241:1
225:μ
221:x
204:μ
195:c
155:]
152:2
149:,
146:0
143:(
23:.
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