Название |
Обозначение |
Параметр |
Носитель |
Плотность вероятности ![{\displaystyle f(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074) |
Функция распределения F(х) |
Характеристическая функция |
Математическое ожидание |
Медиана |
Мода |
Дисперсия |
Коэффициент асимметрии |
Коэффициент эксцесса |
Дифференциальная энтропия |
Производящая функция моментов
|
Равномерное непрерывное |
![{\displaystyle U(a,b)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b29bf0074f2f3e705b7ef9a0946bc0f357961ebe) |
, — коэффициент сдвига, — коэффициент масштаба |
![{\displaystyle [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935) |
![{\displaystyle {\dfrac {1}{b-a}}I\{x\in [a,b]\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f5cf3af1781a097132a99a0faa824b63e0150ff) |
![{\displaystyle {\dfrac {x-a}{b-a}}I\{x\in [a,b]\}+I\{x>b\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b34ee204339a66241f331f91f2c1bf882fd2b8a) |
![{\displaystyle {\dfrac {e^{itb}-e^{ita}}{it(b-a)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd4ff4c83baf223113b4752abf264f05980e1810) |
![{\displaystyle {\frac {a+b}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1325e0aa44cdaf4b2e765a44c7109e6b9ed74e77) |
![{\displaystyle {\frac {a+b}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1325e0aa44cdaf4b2e765a44c7109e6b9ed74e77) |
любое число из отрезка ![{\displaystyle [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935) |
![{\displaystyle {\frac {(b-a)^{2}}{12}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/58f0bcceb78ee2cc3a1c7e62e60115121ff7b3f3) |
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950) |
![{\displaystyle -{\frac {6}{5}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a0fb1dbe6ef983f9fd8af97fcb95a6c38862688) |
![{\displaystyle \ln(b-a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ba6e064a753845b9c858260a231470e537bac09) |
|
Нормальное (гауссовское) |
![{\displaystyle N(\mu ,\sigma ^{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3cbd76720b12f0428a8bf1460b7a67cf2f5f3817) |
— коэффициент сдвига, — коэффициент масштаба |
![{\displaystyle \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc) |
![{\displaystyle {\frac {1}{\sigma {\sqrt {2\pi }}}}\;e^{-{\frac {\left(x-\mu \right)^{2}}{2\sigma ^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc49ee29a10091c7ee605ae56dbde6ad7df5768f) |
![{\displaystyle {\frac {1}{2}}\left(1+\operatorname {erf} \left({\frac {x-\mu }{\sqrt {2\sigma ^{2}}}}\right)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/016caa68c327ccedaf1456a63824dd4d174502d5) |
![{\displaystyle e^{i\,\mu \,t-{\frac {\sigma ^{2}t^{2}}{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28d6ae08e1cb12ec6e567cbbd9832e12be872592) |
![{\displaystyle \mu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161) |
![{\displaystyle \mu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161) |
![{\displaystyle \mu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161) |
![{\displaystyle \sigma ^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5) |
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950) |
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950) |
![{\displaystyle \ln \left(\sigma {\sqrt {2\,\pi \,e}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c47c048d3fbf311a0b8af942f44f02908bec393) |
|
Логнормальное |
![{\displaystyle LN(\mu ,\sigma ^{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0f96ab37f5b4b61c8252853a226e83e9251cc4e) |
![{\displaystyle \mu \in \mathbb {R} ,\sigma >0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a55fbf85eca4c1bd2281cfd4f9fbf4d93dc57f68) |
![{\displaystyle (0;+\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9e5e2bf4d7d4a2c4cd752a29ba2a619cfa89ce7) |
![{\displaystyle {\frac {1}{x\sigma {\sqrt {2\pi }}}}e^{-{\frac {1}{2}}\left({\frac {\ln(x)-\mu }{\sigma }}\right)^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/080e4c779a5aa8a9780fe08ea119f0c7add29325) |
![{\displaystyle {\frac {1}{2}}+{\frac {1}{2}}\mathrm {Erf} \left[{\frac {\ln(x)-\mu }{\sigma {\sqrt {2}}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a9d6e18da5576e1e9e9ebfb992655a16fd71de4) |
![{\displaystyle \sum _{n=0}^{\infty }{\frac {(it)^{n}}{n!}}e^{n\mu +n^{2}\sigma ^{2}/2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6733a3f8e5325d8f77495404c70168cdaa7cfab9) |
![{\displaystyle e^{\mu +\sigma ^{2}/2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de756c5b0f00d3aa53950429a9ec70b5ce76d510) |
![{\displaystyle e^{\mu }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1cd2521c2e2292ade2ad831f116f8d4c036220e) |
![{\displaystyle e^{\mu -\sigma ^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/979e828a58964d542c7ba1ff57e6ac0bcdd4f343) |
![{\displaystyle (e^{\sigma ^{2}}\!\!-1)e^{2\mu +\sigma ^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7cc2f7baa907a3063cc29bbdbdf6fdcfcdade600) |
![{\displaystyle (e^{\sigma ^{2}}\!\!+2){\sqrt {e^{\sigma ^{2}}\!\!-1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68a58ace97758726d2b43d70445f0d4e313de45d) |
![{\displaystyle e^{4\sigma ^{2}}\!\!+2e^{3\sigma ^{2}}\!\!+3e^{2\sigma ^{2}}\!\!-6}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28e1e721fbb8ee78fba51ece9fc9699ebaed237e) |
![{\displaystyle {\frac {1}{2}}+{\frac {1}{2}}\ln(2\pi \sigma ^{2})+\mu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d34493399b3904b64c510044f573fae266683c1) |
|
Гамма-распределение |
![{\displaystyle \Gamma (\alpha ,\beta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/138576f1514e1a60b9994cdb07c7a42c2137d4b9) |
![{\displaystyle \alpha >0,\beta >0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41f5a9ef84eb38b974387eb94d7e16d6f1c9a81c) |
![{\displaystyle \mathbb {R} _{+}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c1f2c2437bae14145e43c54cb7e1ee2701b2106) |
![{\displaystyle {\frac {\alpha ^{\beta }x^{\beta -1}}{\Gamma (\beta )}}e^{-\alpha x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fcdf0e0eedec3502d525d7491bc7311be8c13ace) |
![{\displaystyle {\frac {1}{\Gamma (\beta )}}\gamma (\beta ,\alpha x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/544cce604c15e334672a3cbe9f5463bd6481fe52) |
![{\displaystyle \left(1-{\frac {it}{\alpha }}\right)^{-\beta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65fcf6764e6cf67e1df64156770a09391bf89d0d) |
![{\displaystyle {\frac {\beta }{\alpha }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c7e3e6c309a60c05c3fac1e1de920b7f9e4c674) |
|
при ![{\displaystyle \beta \geq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5ed4a4d3cbac4fa8c095511b86b8a08032c5d17) |
![{\displaystyle {\frac {\beta }{\alpha ^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42cf545d7bcdde7e9bb572ae57e901f910e0c5bb) |
![{\displaystyle {\frac {2}{\sqrt {\beta }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f79cca21516d4923b09fba37352e3f692079f35d) |
![{\displaystyle {\frac {6}{\beta }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba8f103ef2b7dbc99e6cc28a43ff02786e1ca62) |
![{\displaystyle {\begin{aligned}\beta &-\ln \alpha +\ln \Gamma (\beta )\\&+(1-\beta )\psi (\beta )\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9ee8d391ab1a29be5d0f18c280547cf841ed7bd) |
при
|
Экспоненциальное |
![{\displaystyle {\text{Exp}}(\lambda )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41eb7108e974c4852f5af80b305f57161e80f482) |
![{\displaystyle \lambda >0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eea25afc0351140f919cf791c49c1964b8b081de) |
![{\displaystyle \mathbb {R} _{+}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c1f2c2437bae14145e43c54cb7e1ee2701b2106) |
![{\displaystyle \lambda e^{-\lambda x}I\{x>0\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aed486cbd1cb38f329d5f2902583c4c88672f26c) |
![{\displaystyle 1-e^{-\lambda x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6cc79b9dcb7538bec33cfc1dcd3c45b770ca2386) |
![{\displaystyle {\frac {\lambda }{\lambda -it}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/541a4e0a6915e4413d4bab0d9e0352a73407b72c) |
![{\displaystyle {\frac {1}{\lambda }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d496a9397007733fdbbb2f98433e7aab4e51ff4f) |
![{\displaystyle \ln(2)/\lambda }](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfd32abca1be582caa62277f14e2e4556a72131) |
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950) |
![{\displaystyle \lambda ^{-2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f4054ef3203101f810209044d5c3008a80f4759) |
![{\displaystyle 2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f) |
![{\displaystyle 6}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39d81124420a058a7474dfeda48228fb6ee1e253) |
![{\displaystyle 1-\ln(\lambda )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39d6f648643bd3c27df91d4666986e404ccd53a6) |
|
Лапласа |
![{\displaystyle {\text{Laplace}}(\alpha ,\beta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38d6bc5aa5cf02ca18266bef3a7f00d027f66922) |
— коэффициент масштаба, — коэффициент сдвига |
![{\displaystyle \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc) |
![{\displaystyle {\frac {\alpha }{2}}\,e^{-\alpha |x-\beta |}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d81baeddc2637ba3e715db6e71bc15a1b807e7a) |
![{\displaystyle {\begin{cases}{\frac {1}{2}}e^{\alpha (x-\beta )},&x\leqslant \beta \\1-{\frac {1}{2}}e^{-\alpha (x-\beta )},&x>\beta \end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c8dfbdd5ac49bb22d2134fc36c8c2e7f3231183) |
![{\displaystyle {\frac {\alpha ^{2}}{\alpha ^{2}+t^{2}}}e^{it\beta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7267a5aa023a52a7c881008fae0a7483ecd64bca) |
![{\displaystyle \beta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8) |
![{\displaystyle \beta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8) |
![{\displaystyle \beta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8) |
![{\displaystyle {\frac {2}{\alpha ^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e596d2e675032fb37237507c119f5a3a88992b40) |
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950) |
![{\displaystyle 3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/991e33c6e207b12546f15bdfee8b5726eafbbb2f) |
![{\displaystyle \ln {\frac {2e}{\alpha }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/942f46e24b5c06998621310d3264803880ba5c1f) |
|
Коши |
![{\displaystyle {\text{Cauchy}}(x_{0},\gamma )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c643c1122cdf54462f5453f235be28aaf4a3500) |
— коэффициент сдвига, — коэффициент масштаба |
![{\displaystyle \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc) |
![{\displaystyle {1 \over \pi }\left({\gamma \over \gamma ^{2}+(x-x_{0})^{2}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64f8a7dce16bf5731f7d36f75f1ffa6afeadf9dc) |
![{\displaystyle {\frac {1}{\pi }}\mathrm {arctg} \left({\frac {x-x_{0}}{\gamma }}\right)+{\frac {1}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a39df73e695ee89c91ca4ea0d0cbb42251b811c) |
![{\displaystyle e^{i\,x_{0}\,t-\gamma \,|t|}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba0320d95ee5cf1d8c111852bc4481645eb1cd39) |
нет |
![{\displaystyle x_{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf) |
![{\displaystyle x_{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf) |
![{\displaystyle +\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/bddbb0e4420a7e744cf71bd71216e11b0bf88831) |
нет |
нет |
![{\displaystyle \ln(4\,\pi \,\gamma )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56a8bce1a40e6a1387a45421498d7141eabbb5bd) |
нет
|
Бета-распределение |
![{\displaystyle {\text{Beta}}(\alpha ,\beta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0675ff225ee9d8f275e3f7f7fd842f3dff98be29) |
![{\displaystyle \alpha >0,\beta >0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41f5a9ef84eb38b974387eb94d7e16d6f1c9a81c) |
![{\displaystyle [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d) |
![{\displaystyle {\frac {x^{\alpha -1}(1-x)^{\beta -1}}{{\text{B}}(\alpha ,\beta )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc76dd6a8ef4c25e8166f4f8a84171a7a60ad847) |
![{\displaystyle I_{x}(\alpha ,\beta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/663054c3f3dc36c0c8c445386d9e52aea7e26b07) |
![{\displaystyle {}_{1}F_{1}(\alpha ;\alpha +\beta ;i\,t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2404951a0e94d7e5cd53f9f7683f5f3fc3cfa2d1) |
![{\displaystyle {\frac {\alpha }{\alpha +\beta }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28bd51317b477dac57cc5c872c6567a9f4f78396) |
для ![{\displaystyle \alpha ,\beta >1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc3f33fc553c096bb6e12987a13ab58edef863b6) |
для ![{\displaystyle \alpha >1,\beta >1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2f1e017514157062ecc289c4042d17d99a1b77f) |
![{\displaystyle {\frac {\alpha \beta }{(\alpha +\beta )^{2}(\alpha +\beta +1)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/170821a4f691f2daa7afe0a8a8dddceb3ae66d6c) |
![{\displaystyle {\frac {2\,(\beta -\alpha ){\sqrt {\alpha +\beta +1}}}{(\alpha +\beta +2){\sqrt {\alpha \beta }}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43ec71817c032c8eb21b5feadd0ec9b91c747530) |
![{\displaystyle 6\,{\frac {\alpha ^{3}-\alpha ^{2}(2\beta -1)+\beta ^{2}(\beta +1)-2\alpha \beta (\beta +2)}{\alpha \beta (\alpha +\beta +2)(\alpha +\beta +3)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f948e5600ff3f184889dd1d51ea6197aeda24854) |
|
|
хи-квадрат |
![{\displaystyle \chi ^{2}(k)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20d3f0ac864146e53b984408f3c9f75603d0e601) |
— число степеней свободы |
![{\displaystyle \mathbb {R} _{+}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c1f2c2437bae14145e43c54cb7e1ee2701b2106) |
![{\displaystyle {\frac {(1/2)^{k/2}}{\Gamma (k/2)}}x^{k/2-1}e^{-x/2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6f0c4fc9b5936a2746c2689d32718dcf56a9c48) |
![{\displaystyle {\frac {\gamma (k/2,x/2)}{\Gamma (k/2)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b69b4b1161ee5c852a6dff24d5c69ac3f77c2d1) |
![{\displaystyle (1-2\,i\,t)^{-k/2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6f414a9669a01e5b767b6f82e31500877e8c4e7) |
![{\displaystyle k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40) |
примерно ![{\displaystyle k-2/3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40ae3706a97b385e6d9e3b3faa6a600372426fb4) |
если ![{\displaystyle k\geq 2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c797a67c0a51167d373c013a9a020f4568a11754) |
![{\displaystyle 2\,k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d52b32cf63272b9a625b31bd6c48ac49551f6199) |
![{\displaystyle {\sqrt {8/k}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db8e0ba0ca01a74ea24de3e1e51133f256e2c3c7) |
![{\displaystyle 12/k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a69c8e818cdc83079cc238bc28c4c04675d087d4) |
![{\displaystyle {\frac {k}{2}}\!+\!\ln \left[2\Gamma \left({k \over 2}\right)\right]\!+\!\left(1\!-\!{\frac {k}{2}}\right)\psi \left({\frac {k}{2}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68f01393e5f33aee251e935c9fdb24c42d85af2a) |
, если
|
Стьюдента |
![{\displaystyle {\text{t}}(n)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53580cb02459235618e8ae095560a621f8c849d9) |
— число степеней свободы |
![{\displaystyle \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc) |
![{\displaystyle {\frac {\Gamma ({\frac {n+1}{2}})}{{\sqrt {n\pi }}\,\Gamma ({\frac {n}{2}})}}(1+{\frac {x^{2}}{n}})^{-{\frac {n+1}{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47a8f009e8c7dab682e947a5a9c56e7b99e020db) |
![{\displaystyle {\frac {1}{2}}+{x\Gamma \left({\frac {n+1}{2}}\right)}{\frac {\,_{2}F_{1}\left({\frac {1}{2}},{\frac {n+1}{2}};{\frac {3}{2}};-{\frac {x^{2}}{n}}\right)}{{\sqrt {\pi n}}\,\Gamma ({\frac {n}{2}})}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b28d8cb566a332eb70ba2d6ced93634577913fc9) |
для ![{\displaystyle n>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27a6a5d982d54202a14f111cb8a49210501b2c96) |
, если ![{\displaystyle n>1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee74e1cc07e7041edf0fcbd4481f5cd32ad17b64) |
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950) |
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950) |
, если ![{\displaystyle n>2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44e71ac55b9fbf1e9f341b946cda63d61d3ef2cd) |
, если ![{\displaystyle n>3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/257030caae597fd034c2cbcff2cff9dfc4272d20) |
, если ![{\displaystyle n>4}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c6b13dc8b113121cdaf76a723a61aa4f8be1468) |
![{\displaystyle {\begin{matrix}{\frac {n+1}{2}}\left[\psi ({\frac {1+n}{2}})-\psi ({\frac {n}{2}})\right]\\[0.5em]+\log {\left[{\sqrt {n}}B({\frac {n}{2}},{\frac {1}{2}})\right]}\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1cd3fdc61973f68042141bd52ef3b36cd233e8a0) |
Нет
|
Фишера |
![{\displaystyle F(d_{1},d_{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2e1cb0655225361e3eeb28b53e66aab34e74ec2) |
- числа степеней свободы |
![{\displaystyle \mathbb {R} _{+}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c1f2c2437bae14145e43c54cb7e1ee2701b2106) |
![{\displaystyle {\frac {\sqrt {\frac {(d_{1}\,x)^{d_{1}}\,\,d_{2}^{d_{2}}}{(d_{1}\,x+d_{2})^{d_{1}+d_{2}}}}}{x\,\mathrm {B} \!\left({\frac {d_{1}}{2}},{\frac {d_{2}}{2}}\right)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b92202ea02118bb75d06208c9e556cb222972bd0) |
![{\displaystyle I_{\frac {d_{1}x}{d_{1}x+d_{2}}}(d_{1}/2,d_{2}/2)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d79f7786b87e45ac16afd1e5f9548adc33a384a) |
![{\displaystyle {\frac {\Gamma ({\frac {d_{1}+d_{2}}{2}})}{\Gamma ({\tfrac {d_{2}}{2}})}}U\!\left({\frac {d_{1}}{2}},1-{\frac {d_{2}}{2}},-{\frac {d_{2}}{d_{1}}}\imath s\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c6565a4f32e7561e48c8149ade5eb7b17fea21b) |
, если ![{\displaystyle d_{2}>2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8becc7f9a26666a2faee158c209dba7b42c4b66) |
|
, если ![{\displaystyle d_{1}>2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e308c920cd72c86c6a1a8f84b0eadc3a807a711f) |
если ![{\displaystyle d_{2}>4}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1102915ed0508d2dc5afe5bd440e7dfad0249887) |
![{\displaystyle {\frac {(2d_{1}+d_{2}-2){\sqrt {8(d_{2}-4)}}}{(d_{2}-6){\sqrt {d_{1}(d_{1}+d_{2}-2)}}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6a345df85b068e9decf9829a4b0f9d35327de77) если ![{\displaystyle d_{2}>6}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cde6f4f94cb60b4cc472d49a8f614fe723590be) |
![{\displaystyle 12{\frac {d_{1}(5d_{2}-22)(d_{1}+d_{2}-2)+(d_{2}-4)(d_{2}-2)^{2}}{d_{1}(d_{2}-6)(d_{2}-8)(d_{1}+d_{2}-2)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/030fe52f743c19b86c0504353aba762898463c71) |
![{\displaystyle \ln \Gamma \left({\tfrac {d_{1}}{2}}\right)+\ln \Gamma \left({\tfrac {d_{2}}{2}}\right)-\ln \Gamma \left({\tfrac {d_{1}+d_{2}}{2}}\right)+\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a507d266e4e8fc6eeecff472ac0cd3c55b8ed681)
![{\displaystyle \left(1-{\tfrac {d_{1}}{2}}\right)\psi \left(1+{\tfrac {d_{1}}{2}}\right)-\left(1+{\tfrac {d_{2}}{2}}\right)\psi \left(1+{\tfrac {d_{2}}{2}}\right)\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f87cb3fbf968cbc40c36c51e6cc777bdfcfacb93)
|
Рэлея |
![{\displaystyle \mathrm {Rayleigh} (\sigma )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aba7d3595625f3a024b8a1ad806683a64f5c7ecc) |
![{\displaystyle \sigma }](https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36) |
![{\displaystyle \mathbb {R} _{+}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c1f2c2437bae14145e43c54cb7e1ee2701b2106) |
![{\displaystyle {\frac {x}{{\sigma }^{2}}}e^{-{\frac {{x}^{2}}{2{{\sigma }^{2}}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66c5d299a01402a9c8baf5168c5e5b89dcdba2f4) |
![{\displaystyle 1-e^{\frac {-x^{2}}{2\sigma ^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06294eee9511cdf796b341576400df3f7c6224ab) |
![{\displaystyle 1\!-\!\sigma te^{-\sigma ^{2}t^{2}/2}{\sqrt {\frac {\pi }{2}}}\!\left({\textrm {erfi}}\!\left({\frac {\sigma t}{\sqrt {2}}}\right)\!-\!i\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/264980c5948431356e42d3b9f5785e99e31fedac) |
![{\displaystyle {\sqrt {\frac {\pi }{2}}}\sigma }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3083f47d79817f2be95f22a9973206fcd7275506) |
![{\displaystyle \sigma {\sqrt {\ln(4)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad5e52f6a89d6cf17ab27e52de02bff4cc825632) |
![{\displaystyle \sigma }](https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36) |
![{\displaystyle \left(2-\pi /2\right){{\sigma }^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4e350238f82cf601c5b76020230342daa25d304) |
![{\displaystyle {\frac {2{\sqrt {\pi }}(\pi -3)}{(4-\pi )^{3/2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7abc398e5e3dc51f2cb47e9402dde1d61020beb) |
![{\displaystyle -{\frac {6\pi ^{2}-24\pi +16}{(4-\pi )^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1bcd42ad8e7db81c5a4c00ec979c2991dcad8372) |
![{\displaystyle 1+\ln \left({\frac {\sigma }{\sqrt {2}}}\right)+{\frac {\gamma }{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/074c0bc67dbb029023939280ba79436321c4fece) |
|
Вейбулла |
![{\displaystyle \mathrm {W} (k,\lambda )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f632256adfc19c9d44caa91455f4038e0c9bb35) |
- коэффициент масштаба, - коэффициент формы |
![{\displaystyle \mathbb {R} _{+}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c1f2c2437bae14145e43c54cb7e1ee2701b2106) |
![{\displaystyle {\frac {k}{\lambda }}\left({\frac {x}{\lambda }}\right)^{k-1}e^{-\left({\frac {x}{\lambda }}\right)^{k}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b76823f75d8d484ff652dc9ad98cd8dee172f5f3) |
![{\displaystyle 1-e^{-\left({\frac {x}{\lambda }}\right)^{k}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9e03950f75b5f19b11f31d6ef098edb007ff2a8) |
![{\displaystyle \sum _{n=0}^{\infty }{\frac {(it)^{n}\lambda ^{n}}{n!}}\Gamma (1+n/k)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2e10044b884683e5083847ccb8aa3df64f990b2) |
![{\displaystyle \lambda \Gamma \left(1+{\frac {1}{k}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ccb6439a38b9b2de5d1369ff828fe889cd43389b) |
![{\displaystyle \lambda \ln(2)^{1/k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/229e60de0d9e1153df076e22aac2b3965ad3a313) |
для ![{\displaystyle k>1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5cda43bd4034dc2d04cd562005d0af81d3d2dbc6) |
![{\displaystyle \lambda ^{2}\Gamma \left(1+{\frac {2}{k}}\right)-\mu ^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5b23d066928cea324d7af31eac18d5a41abef29) |
![{\displaystyle {\frac {\Gamma (1+{\frac {3}{k}})\lambda ^{3}-3\mu \Gamma (1+{\frac {2}{k}})\lambda ^{2}+2\mu ^{3}}{\sigma ^{3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bccb795618c853fbd8f48c2bdb937b9de69c80e) |
![{\displaystyle {\frac {\lambda ^{4}\Gamma \left(1+{\frac {4}{k}}\right)-4\lambda ^{3}\mu \Gamma \left(1+{\frac {3}{k}}\right)+6\lambda ^{2}\mu ^{2}\Gamma \left(1+{\frac {2}{k}}\right)-3\mu ^{4}}{\sigma ^{4}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5035b74e37847613750d2cdf48a1fa4c60840bc0) |
![{\displaystyle \gamma \left(1\!-\!{\frac {1}{k}}\right)+\left({\frac {\lambda }{k}}\right)^{k}+\ln \left({\frac {\lambda }{k}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ce7c1055685df2d9c04e9b9ecf4c92dec9216d7) |
|
Логистическое |
![{\displaystyle L(\mu ,s)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5396007122aa3b32fdf8ad76cabcd2b15aecac3d) |
, ![{\displaystyle s>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76beea94b6662bd490c61c0628dddd8a8cd35538) |
![{\displaystyle \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc) |
![{\displaystyle {\frac {e^{-(x-\mu )/s}}{s\left(1+e^{-(x-\mu )/s}\right)^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d42088b22c0c7577eb7a68d827061870368d054) |
![{\displaystyle {\frac {1}{1+e^{-(x-\mu )/s}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/58e537816cff6b1c914d540655fee0c7c8091c5c) |
для ![{\displaystyle |ist|<1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7daa20eec94bc5ef6b0e5c7529c226446c336af) |
![{\displaystyle \mu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161) |
![{\displaystyle \mu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161) |
![{\displaystyle \mu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161) |
![{\displaystyle {\frac {\pi ^{2}}{3}}s^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88df3819d9137593623be5ba54e0289427bc0f1b) |
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950) |
![{\displaystyle 6/5}](https://wikimedia.org/api/rest_v1/media/math/render/svg/927a3fc853d2c544bcb48b4404bb7c0fbbbc6b4b) |
![{\displaystyle \ln(s)+2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49fb1821785a0c476c37a54e86119600830b7b80) |
![{\displaystyle e^{\mu \,t}\,\mathrm {B} (1-s\,t,\;1+s\,t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f572c51cc52bc3e8a5919b518ba9b344de487dba) для
|
Вигнера |
![{\displaystyle \rho (R)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0761345b878fb29f4c2481c267ef2aa61c2a1ab) |
- радиус |
![{\displaystyle [-R;+R]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1220f1c83fabcfbacecc48e24894de39f2457a44) |
![{\displaystyle {\frac {2}{\pi R^{2}}}\,{\sqrt {R^{2}-x^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9deb07bde8b4427a2bd4ac7e15c6e49f6a18fd53) |
для ![{\displaystyle -R\leq x\leq R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7486e1e1b1f66d294905ad9f650b91ae0bda6d) |
![{\displaystyle 2\,{\frac {J_{1}(R\,t)}{R\,t}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/975ad34148e8b5e9d8c2916bb4d68f279e2ee3b9) |
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950) |
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950) |
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950) |
![{\displaystyle {\frac {R^{2}}{4}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f49f5b99eab6a0cbdb24d34a898ba6e1903fccc) |
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950) |
![{\displaystyle -1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac) |
![{\displaystyle \ln(\pi R)-{\frac {1}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de41881d5e9eece4ea5f64daf8efb55db6545fc9) |
|
Парето |
![{\displaystyle {\text{Pareto}}(k,x_{\text{m}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5557eee88a55a5111d4bcf7e1b98964230fb9dc7) |
— коэффициент масштаба, ![{\displaystyle k>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27b3af208b148139eefc03f0f80fa94c38c5af45) |
![{\displaystyle [x_{\text{m}};+\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84e58663e79bc99757a73087bc004a3cafaa4a68) |
![{\displaystyle {\frac {k\,x_{\text{m}}^{k}}{x^{k+1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd360dbb5e0488d635dde3e897c79ac381f3db30) |
![{\displaystyle 1-\left({\frac {x_{\text{m}}}{x}}\right)^{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36a21bd5c05422f5d22485a4e788433138ed5a22) |
![{\displaystyle k{\big (}\Gamma (-k){\big [}x_{\text{m}}^{k}(-it)^{k}-(-ix_{\text{m}}t)^{k}{\big ]}+E_{\text{k+1}}(-ix_{\text{m}}t){\big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82b2a7e6f9af7758fd69d56b96a670713e8e1473) |
, если ![{\displaystyle k>1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5cda43bd4034dc2d04cd562005d0af81d3d2dbc6) |
![{\displaystyle x_{\text{m}}{\sqrt[{k}]{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4e35961fe9242e183d968bd97508d7871d93464) |
![{\displaystyle x_{\text{m}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46d897493a7617e16afb4749af96db61c9e6ce23) |
при ![{\displaystyle k>2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa7a4bb0e17a911cb4b3f6c4e455be472a295299) |
при ![{\displaystyle k>3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45c55b4ff0d61c81d264463917a795a521e8e5c8) |
при ![{\displaystyle k>4}](https://wikimedia.org/api/rest_v1/media/math/render/svg/687a26890d840046dc0cbdf7ec3a280110b4d0a7) |
![{\displaystyle \ln \left({\frac {k}{x_{\text{m}}}}\right)-{\frac {1}{k}}-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7d7017e90811c3d04150259e35d6f96b7a1b691) |
нет
|