(統計)Gamma ($\Gamma$) distribution，伽瑪分布

 

Gamma ($\Gamma$) distribution，伽瑪分布

在一個滿足poisson過程實驗，$X(r.v.)$為直到$\alpha$次成功之時間。每單位時間內成功平均次數$\lambda$。(每平均\beta時間成功1次)

$\alpha >0$    $\lambda>0$

$\Gamma(\alpha,\lambda)=f_X(x)=\left\{\begin{array}{l}\cfrac{\lambda^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\lambda x}\ \ &，0<x<\infty \\0\ \ &，o.w \end{array}\right.$

$\alpha >0$    $\beta>0$

$\Gamma(\alpha,\beta)=f_X(x)=\left\{\begin{array}{l}\cfrac{1}{\Gamma(\alpha)\beta^{\alpha}}x^{\alpha-1}e^{-\frac{x}{\beta}}\ \ &，0<x<\infty \\0\ \ &，o.w \end{array}\right.$

E(X),V(X)

$E(X)=\cfrac{\alpha}{\lambda}=\alpha \beta$

$V(X)=\cfrac{\alpha^2}{\lambda}=\alpha \beta^2$

m.g.f.

$M(t)=\cfrac{1}{({1-\beta })^{\alpha}}$，  $t<\cfrac{1}{\beta}$

伽瑪分配和卡方分配隨機變數之變數變換

若$X\sim \Gamma(\alpha,\lambda) \implies Y=2\lambda X\sim \chi^2_{2\alpha}$

若 $X\sim \chi^2_r \implies Y=\cfrac{aX}{b}\sim \Gamma(\cfrac{r}{2},\lambda=\cfrac{1}{2 \frac{a}{b}}) $