指数分布

参数 概率密度函數 累積分布函數 ${\displaystyle \lambda >0\,}$ 率 ${\displaystyle x\in [0;\infty )\!}$ ${\displaystyle \,\lambda e^{-\lambda x}}$ ${\displaystyle 1-e^{-\lambda x}}$ ${\displaystyle \lambda ^{-1}\,}$ ${\displaystyle \ln(2)/\lambda \,}$ ${\displaystyle 0\,}$ ${\displaystyle \lambda ^{-2}\,}$ ${\displaystyle 2\,}$ ${\displaystyle 6\,}$ ${\displaystyle 1-\ln(\lambda )\,}$ ${\displaystyle \left(1-{\frac {t}{\lambda }}\right)^{-1}\,}$ ${\displaystyle \left(1-{\frac {it}{\lambda }}\right)^{-1}\,}$

記號

${\displaystyle X\sim {\text{Exp}}(\lambda )}$${\displaystyle X\sim {\text{Exp}}(\beta )}$

${\displaystyle f(x;{\color {Red}\lambda })=\left\{{\begin{matrix}{\color {Red}\lambda }e^{-{\color {Red}\lambda }x}&x\geq 0,\\0&,\;x<0.\end{matrix}}\right.}$

${\displaystyle f(x;{\color {Red}\beta })=\left\{{\begin{matrix}{\color {Red}{\frac {1}{\beta }}}e^{-{\color {Red}{\frac {1}{\beta }}}x}&x\geq 0,\\0&,\;x<0.\end{matrix}}\right.}$

${\displaystyle F(x;{\color {Red}\lambda })=\left\{{\begin{matrix}1-e^{-\color {Red}{\lambda }x}&,\;x\geq 0,\\0&,\;x<0.\end{matrix}}\right.}$

${\displaystyle F(x;{\color {Red}\beta })=\left\{{\begin{matrix}1-e^{-{\color {Red}{\frac {1}{\beta }}}x}&,\;x\geq 0,\\0&,\;x<0.\end{matrix}}\right.}$

特性

期望值与變異數

${\displaystyle \mathbf {E} (X)={\frac {1}{\color {Red}{\lambda }}}={\color {Red}\beta }}$

X方差是：

${\displaystyle \mathbf {Var} (X)={\frac {1}{\color {Red}{\lambda ^{2}}}}={\color {Red}\beta ^{2}}}$

X偏態系数是： V[X] = 1

无记忆性

${\displaystyle P(T>s+t\;|\;T>t)=P(T>s)\;\;{\hbox{for all}}\ s,t\geq 0.}$

与泊松过程的关系

${\displaystyle {\frac {e^{-\lambda t}(\lambda t)^{0}}{0!}}=e^{-\lambda t}}$,

四分位数

${\displaystyle F^{-1}(p;\lambda )={\frac {-\ln(1-p)}{\lambda }},\qquad 0\leq p<1}$
• 第一四分位数：${\displaystyle \ln(4/3)/\lambda \,}$
• 中位数${\displaystyle \ln(2)/\lambda \,}$
• 第三四分位数：${\displaystyle \ln(4)/\lambda \,}$

参数估计

最大概似法

${\displaystyle L(\lambda )=\prod _{i=1}^{n}\lambda \,\exp(-\lambda x_{i})=\lambda ^{n}\,\exp \!\left(\!-\lambda \sum _{i=1}^{n}x_{i}\right)=\lambda ^{n}\exp \left(-\lambda n{\overline {x}}\right)}$

${\displaystyle {\overline {x}}={1 \over n}\sum _{i=1}^{n}x_{i}}$是样本期望値。

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} \lambda }}\ln L(\lambda )={\frac {\mathrm {d} }{\mathrm {d} \lambda }}\left(n\ln(\lambda )-\lambda n{\overline {x}}\right)={n \over \lambda }-n{\overline {x}}\ \left\{{\begin{matrix}>0&{\mbox{if}}\ 0<\lambda <1/{\overline {x}},\\\\=0&{\mbox{if}}\ \lambda =1/{\overline {x}},\\\\<0&{\mbox{if}}\ \lambda >1/{\overline {x}}.\end{matrix}}\right.}$

${\displaystyle {\widehat {\lambda }}={\frac {1}{\overline {x}}}}$

参考文獻

1. Donald E. Knuth (1998). The Art of Computer Programming, volume 2: Seminumerical Algorithms, 3rd edn. Boston: Addison-Wesley. ISBN 0-201-89684-2. pp. 133
2. Luc Devroye (1986). Non-Uniform Random Variate Generation. New York: Springer-Verlag. ISBN 0-387-96305-7. pp. 392–401