伽玛分布(英语:Gamma distribution)是统计学的一种连续概率分布。伽玛分布中的参数α,称为形状参数,β称为尺度参数。 事实速览 参数, 值域 ...Gamma 概率密度函数 累积分布函数参数 k > 0 {\displaystyle k>0\,} 形状参数 (实数) θ > 0 {\displaystyle \theta >0\,} 尺度参数 (实数)值域 x ∈ ( 0 ; ∞ ) {\displaystyle x\in (0;\infty )\!} 概率密度函数 x k − 1 exp ( − x / θ ) Γ ( k ) θ k {\displaystyle x^{k-1}{\frac {\exp {\left(-x/\theta \right)}}{\Gamma (k)\,\theta ^{k}}}\,\!} 累积分布函数 γ ( k , x / θ ) Γ ( k ) {\displaystyle {\frac {\gamma (k,x/\theta )}{\Gamma (k)}}\,\!} 期望 k θ {\displaystyle k\theta \,\!} 中位数 no simple closed form众数 ( k − 1 ) θ {\displaystyle (k-1)\theta \,\!} for k ≥ 1 {\displaystyle k\geq 1\,\!} 方差 k θ 2 {\displaystyle k\theta ^{2}\,\!} 偏度 2 k {\displaystyle {\frac {2}{\sqrt {k}}}\,\!} 峰度 6 k {\displaystyle {\frac {6}{k}}\,\!} 熵 k + ln θ + ln Γ ( k ) {\displaystyle k+\ln \theta +\ln \Gamma (k)\!} + ( 1 − k ) ψ ( k ) {\displaystyle +(1-k)\psi (k)\!} 矩生成函数 ( 1 − θ t ) − k {\displaystyle (1-\theta \,t)^{-k}\,\!} for t < 1 / θ {\displaystyle t<1/\theta \,\!} 特征函数 ( 1 − θ i t ) − k {\displaystyle (1-\theta \,i\,t)^{-k}\,\!} 关闭 实验定义与观念 假设X1, X2, ... Xn 为连续发生事件的等候时间,且这n次等候时间为独立的,那么这n次等候时间之和Y (Y=X1+X2+...+Xn)服从伽玛分布,即 Y~Gamma(α , β),亦可记作Y~Gamma(α , λ),其中α = n,而 β 与λ互为倒数关系,λ 表单位时间内事件的发生率。 指数分布为α = 1的伽玛分布。 记号 有两种表记方法: X ∼ Γ ( α , β ) {\displaystyle X\sim \Gamma (\alpha ,\beta )} 或 X ∼ Γ ( α , λ ) {\displaystyle X\sim \Gamma (\alpha ,\lambda )} 两者所表达意义相同,只要将以下式子做 λ = 1 β {\displaystyle {\color {Red}\lambda ={\frac {1}{\beta }}}} 的替换即可,即,其概率密度函数为: f ( x ) = x ( α − 1 ) λ α e ( − λ x ) Γ ( α ) = x ( α − 1 ) e ( − 1 β x ) β α Γ ( α ) {\displaystyle f\left(x\right)={\frac {x^{\left(\alpha -1\right)}{\color {Red}\lambda }^{\alpha }e^{\left(-{\color {Red}\lambda }x\right)}}{\Gamma \left(\alpha \right)}}={\frac {x^{\left(\alpha -1\right)}e^{\left(-{\color {Red}{\frac {1}{\beta }}}x\right)}}{{\color {Red}\beta }^{\alpha }\Gamma \left(\alpha \right)}}} ,x > 0 其中Gamma函数之特征为: { Γ ( α ) = ( α − 1 ) ! if α is Z + Γ ( α ) = ( α − 1 ) Γ ( α − 1 ) if α is R + Γ ( 1 2 ) = π {\displaystyle {\begin{cases}\Gamma (\alpha )=(\alpha -1)!&{\mbox{if }}\alpha {\mbox{ is }}\mathbb {Z} ^{+}\\\Gamma (\alpha )=(\alpha -1)\Gamma (\alpha -1)&{\mbox{if }}\alpha {\mbox{ is }}\mathbb {R} ^{+}\\\Gamma \left({\frac {1}{2}}\right)={\sqrt {\pi }}\end{cases}}} 特性 母函数、期望、方差 Gamma分配的矩母函数(m.g.f) M x ( t ) = E ( e x t ) = λ α Γ ( α ) ∫ 0 ∞ e x t x α − 1 e − λ x d x = ( λ λ − t ) α = ( 1 − β t ) − α {\displaystyle M_{x}\left(t\right)=E\left(e^{xt}\right)={\frac {\lambda ^{\alpha }}{\Gamma \left(\alpha \right)}}\int _{0}^{\infty }e^{xt}x^{\alpha -1}e^{-\lambda x}dx=\left({\frac {\lambda }{\lambda -t}}\right)^{\alpha }=\left(1-{\beta }{t}\right)^{-\alpha }} 概率母函数(p.g.f) K x ( t ) = ln M x ( t ) = α [ ln λ − ln ( λ − t ) ] {\displaystyle K_{x}\left(t\right)=\ln M_{x}\left(t\right)=\alpha \left[\ln \lambda -\ln \left(\lambda -t\right)\right]} 期望 d K x ( t ) d t = α λ − t , w h e n ( t = 0 ) , E ( X ) = α λ = α β {\displaystyle {\frac {dK_{x}\left(t\right)}{dt}}={\frac {\alpha }{\lambda -t}},\quad when(t=0),E\left(X\right)={\frac {\alpha }{\color {Red}\lambda }}=\alpha {\color {Red}\beta }} 方差 d 2 K x ( t ) d t 2 = α ( λ − t ) 2 , w h e n ( t = 0 ) , σ 2 ( X ) = α λ 2 = α β 2 {\displaystyle {\frac {d^{2}K_{x}\left(t\right)}{dt^{2}}}={\frac {\alpha }{\left(\lambda -t\right)^{2}}},\quad when(t=0),\sigma ^{2}\left(X\right)={\frac {\alpha }{\color {Red}{\lambda ^{2}}}}=\alpha {\color {Red}{\beta ^{2}}}} Gamma的可加性 当两随机变量服从Gamma分布,且相互独立,且参数( λ {\displaystyle \lambda } 或 β {\displaystyle \beta } )相同时,Gamma分布具有可加性。 ∐ { r . v . X ∼ Γ ( α 1 , λ ) r . v . Y ∼ Γ ( α 2 , λ ) ⟹ X + Y ∼ Γ ( α 1 + α 2 , λ ) {\displaystyle \coprod {\begin{cases}r.v.X\sim \Gamma \left({\color {green}\alpha _{1}},\lambda \right)\\r.v.Y\sim \Gamma \left({\color {green}\alpha _{2}},\lambda \right)\end{cases}}\Longrightarrow X+Y\sim \Gamma \left({\color {green}\alpha _{1}+\alpha _{2}},\lambda \right)} 外部链接 LDA-math-神奇的Gamma函数(页面存档备份,存于互联网档案馆) 分布计算器(页面存档备份,存于互联网档案馆)(英文) Wikiwand - on Seamless Wikipedia browsing. On steroids.