Gamma 概率密度函數 ![Probability density plots of gamma distributions](//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Gamma_distribution_pdf.png/325px-Gamma_distribution_pdf.png) |
累積分布函數 ![Cumulative distribution plots of gamma distributions](//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Gamma_distribution_cdf.png/325px-Gamma_distribution_cdf.png) |
参数 | shape (real)
scale (real) |
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值域 | ![{\displaystyle x\in (0;\infty )\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0ac3b4aa4e92d95eabc77be7aa19c29d55343b2) |
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概率密度函数 | ![{\displaystyle x^{k-1}{\frac {\exp {\left(-x/\theta \right)}}{\Gamma (k)\,\theta ^{k}}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/172af602ae7be6663d8f58058b6e1cab67c73611) |
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累積分布函數 | ![{\displaystyle {\frac {\gamma (k,x/\theta )}{\Gamma (k)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80be250b1603368e1a5bd74e211c734d20bcb784) |
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期望值 | ![{\displaystyle k\theta \,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09d992af91e9bb082caf60bcbee2a0f7a4c73033) |
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中位數 | no simple closed form |
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眾數 | for ![{\displaystyle k\geq 1\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76f945852396fbfa7f0a37fd80bdee1c5e1bfbf1) |
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方差 | ![{\displaystyle k\theta ^{2}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ad5d5e6879a48782a7e7f9026e82884631d4804) |
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偏度 | ![{\displaystyle {\frac {2}{\sqrt {k}}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2b9e3e0be057fa12cf1b4c42b32261622650c3d) |
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峰度 | ![{\displaystyle {\frac {6}{k}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/950ff63899a6fd32c59ea615c2073a69d1b0aa33) |
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熵 | ![{\displaystyle k+\ln \theta +\ln \Gamma (k)\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ae69f4ec6d3900d92570f1da9b130d23b712dfb)
![{\displaystyle +(1-k)\psi (k)\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d4c575d926dff7d9f1901cca915a9a80db28dbe) |
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矩生成函数 | for ![{\displaystyle t<1/\theta \,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf4326a62e7872c59e63a22af5b75af53c3a406b) |
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特徵函数 | ![{\displaystyle (1-\theta \,i\,t)^{-k}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7e3f419b0064682162332e1fb6f01fe70f9d663) |
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伽玛分布(英語:Gamma distribution)是統計學的一種連續機率分布。伽玛分佈中的母數α,稱為形狀参数,β稱為尺度参数。
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),亦可記作Y~Gamma(α , λ),其中α = n,而 β 與λ互為倒數關係,λ 表單位時間內事件的發生率。
指数分布為α = 1的伽瑪分布。
有兩種表記方法:
或
兩者所表達意義相同,只要將以下式子做
的替換即可,即,其機率密度函數為:
,x > 0
其中Gamma函数之特徵為:
![{\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 }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6472b9755975d8444c8a5e6d2000fddf4288de1b)
![{\displaystyle K_{x}\left(t\right)=\ln M_{x}\left(t\right)=\alpha \left[\ln \lambda -\ln \left(\lambda -t\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f3c9932ee15d43b2588004f15ca6294b05b4462)
![{\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 }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7044e8a02fc68cefa405fb52dc9fb0223faced3)
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3f01d930bbf89597598d9d3966e79a237cd4950)
當兩隨機變數服從Gamma分布,且相互獨立,且母數(
或
)相同時,Gamma分布具有可加性。
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/755c074033ab2e1b0d638e69f35d9b5302ee53ac)