# 甘別分佈

變數 概率密度函數 累積分佈函數 ${\displaystyle \mu ,}$ location (real)${\displaystyle \beta >0,}$ scale (real) ${\displaystyle x\in \mathbb {R} }$ ${\displaystyle {\frac {1}{\beta }}e^{(z-e^{z})}}$ where ${\displaystyle z={\frac {x-\mu }{\beta }}}$ ${\displaystyle e^{-e^{-(x-\mu )/\beta }}}$ ${\displaystyle \mu +\beta \gamma }$ where ${\displaystyle \gamma }$ is the Euler–Mascheroni constant ${\displaystyle \mu -\beta \ln(\ln 2)}$ ${\displaystyle \mu }$ ${\displaystyle {\frac {\pi ^{2}}{6}}\beta ^{2}}$ ${\displaystyle {\frac {12{\sqrt {6}}\,\zeta (3)}{\pi ^{3}}}\approx 1.14}$ ${\displaystyle {\frac {12}{5}}}$ ${\displaystyle \ln(\beta )+\gamma +1}$ ${\displaystyle \Gamma (1-\beta t)e^{\mu t}}$ ${\displaystyle \Gamma (1-i\beta t)e^{i\mu t}}$

## 定義

${\displaystyle F(x;\mu ,\beta )=e^{-e^{-(x-\mu )/\beta }}}$

### 標準甘別分佈

${\displaystyle F(x)=e^{-e^{(-x)}}\,}$

${\displaystyle f(x)=e^{-(x+e^{-x})}}$

${\displaystyle \kappa _{n}=(n-1)!\zeta (n)}$

## 特性

${\displaystyle \operatorname {E} (X)=\mu +\gamma \beta }$

## 相關分佈

• 若果${\displaystyle X}$ 具有甘別分佈，係噉Y= − X嘅條件分佈具有甘佩茲分佈（假設Y係正，或者等效噉假設X係負）。Y嘅 cdf GFX嘅 cdf）有拏褦，對於y > 0有公式：${\displaystyle G(y)=P(Y\leq y)=P(X\geq -y|X\leq 0)=(F(0)-F(-y))/F(0)}$ ；所以，密度同${\displaystyle g(y)=f(-y)/F(0)}$ 相關：甘佩茲密度同反射嘅甘別密度成正比，僅衹限於正半線。[4]
• 若果${\displaystyle X}$ 指數分佈、變數均值係 1 嘅，係噉− log( X ) 具有標準甘別分佈。
• 若果${\displaystyle X\sim \mathrm {Gumbel} (\alpha _{X},\beta )}$ ${\displaystyle Y\sim \mathrm {Gumbel} (\alpha _{Y},\beta )}$ 係獨立嘅，係噉${\displaystyle X-Y}$ 邏輯分佈${\displaystyle X-Y\sim \mathrm {Logistic} (\alpha _{X}-\alpha _{Y},\beta )\,}$
• 若果${\displaystyle X,Y\sim \mathrm {Gumbel} (\alpha ,\beta )}$ 係獨立嘅，係噉${\displaystyle X+Y\nsim \mathrm {Logistic} (2\alpha ,\beta )}$ 。注意${\displaystyle E(X+Y)=2\alpha +2\beta \gamma \neq 2\alpha =E\left(\mathrm {Logistic} (2\alpha ,\beta )\right)}$

## 計法

### 概率論文

${\displaystyle -\ln[-\ln(F)]=(x-\mu )/\beta }$

### 生成甘別變數

${\displaystyle Q(p)=\mu -\beta \ln(-\ln(p))}$

## 考

1. Gumbel, E.J. (1935), "Les valeurs extrêmes des distributions statistiques" (PDF), Annales de l'Institut Henri Poincaré, 5 (2): 115–158
2. Gumbel E.J. (1941). "The return period of flood flows". The Annals of Mathematical Statistics, 12, 163–190.
3. Oosterbaan, R.J. (1994). "Chapter 6 Frequency and Regression Analysis" (PDF). 出自 Ritzema, H.P. (編). Drainage Principles and Applications, Publication 16. Wageningen, The Netherlands: International Institute for Land Reclamation and Improvement (ILRI). pp. 175–224. ISBN 90-70754-33-9.
4. Willemse, W.J.; Kaas, R. (2007). "Rational reconstruction of frailty-based mortality models by a generalisation of Gompertz' law of mortality" (PDF). Insurance: Mathematics and Economics. 40 (3): 468. doi:10.1016/j.insmatheco.2006.07.003. 原著 (PDF)喺2017年8月9號歸檔. 喺2021年7月4號搵到.
5. CumFreq, software for probability distribution fitting
6. [https://math.stackexchange.com/questions/3527556/gumbel-distribution-and-exponential-distribution?noredirect=1#comment7669633_3527556 user49229, Gumbel distribution and exponential distribution ]
7. Gumbel, E.J. (1954). Statistical theory of extreme values and some practical applications. Applied Mathematics Series.第33卷 (第1版). U.S. Department of Commerce, National Bureau of Standards. ASIN B0007DSHG4.
8. Burke, Eleanor J.; Perry, Richard H.J.; Brown, Simon J. (2010). "An extreme value analysis of UK drought and projections of change in the future". Journal of Hydrology. 388 (1–2): 131–143. Bibcode:2010JHyd..388..131B. doi:10.1016/j.jhydrol.2010.04.035.
9. Erdös, Paul; Lehner, Joseph (1941). "The distribution of the number of summands in the partitions of a positive integer". Duke Mathematical Journal. 8 (2): 335. doi:10.1215/S0012-7094-41-00826-8.
10. Kourbatov, A. (2013). "Maximal gaps between prime k-tuples: a statistical approach". Journal of Integer Sequences. 16. arXiv:1301.2242. Bibcode:2013arXiv1301.2242K. Article 13.5.2.
11. Adams, Ryan. "The Gumbel-Max Trick for Discrete Distributions".