ガンマ分布

tags: probability-theory

確率密度関数

$$f\left(x\right)=\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}\frac{1}{\beta }{\left(\frac{x}{\beta }\right)}^{\alpha -1}{e}^{-x/\beta }=\frac{x^{\alpha -1}}{{\beta }^{\alpha }\mathrm{\Gamma }\left(\alpha \right)}\exp (-\frac{x}{\beta })$$

確率密度関数の形状

ガンマ分布の密度関数は、

$x\to \mathrm{\infty }$で

$0$に収束する。

$$\underset{x\to \mathrm{\infty }}{lim}f\left(x\right)=0$$
また

$x\to 0$の挙動は

$\alpha$の

$1$との大小関係で定まる。

$$\underset{x\to 0}{lim}f\left(x\right)=\{\begin{array}{ll}0& \alpha >1\\ \mathrm{\infty }& \alpha <1\end{array}$$

密度関数の導関数

$$\begin{array}{rl}\frac{\partial }{\partial x}f\left(x\right)& =\frac{(\alpha -1)x^{\alpha -2}}{{\beta }^{\alpha }\mathrm{\Gamma }\left(\alpha \right)}\exp (-\frac{x}{\beta })-\frac{x^{\alpha -1}}{{\beta }^{\alpha +1}\mathrm{\Gamma }\left(\alpha \right)}\exp (-\frac{x}{\beta })\\ & =\frac{x^{\alpha -2}}{{\beta }^{\alpha }\mathrm{\Gamma }\left(\alpha \right)}\exp (-\frac{x}{\beta })\times (\alpha -1-\frac{x}{\beta })\end{array}$$
は

$\alpha \le 1$ならば常に負、

$\alpha >1$ならば

$x=(\alpha -1)\beta$で符号が正から負に変わる。

平均

$$E\left[X\right]=\alpha \beta$$

$$\begin{array}{rl}E\left[X\right]& ={\int }_0^{\mathrm{\infty }}x\frac{x^{\alpha -1}}{{\beta }^{\alpha }\mathrm{\Gamma }\left(\alpha \right)}\exp (-\frac{x}{\beta })dx\\ & ={\int }_0^{\mathrm{\infty }}\frac{x^{\alpha }}{{\beta }^{\alpha }\mathrm{\Gamma }\left(\alpha \right)}\exp (-\frac{x}{\beta })dx\\ & ={\int }_0^{\mathrm{\infty }}\frac{\alpha \beta }{\beta }\frac{x^{\alpha }}{{\beta }^{\alpha }\alpha \mathrm{\Gamma }\left(\alpha \right)}\exp (-\frac{x}{\beta })dx\\ & =\alpha \beta {\int }_0^{\mathrm{\infty }}\frac{x^{\alpha }}{{\beta }^{\alpha +1}\mathrm{\Gamma }(\alpha +1)}\exp (-\frac{x}{\beta })dx\\ & =\alpha \beta \times 1\\ & =\alpha \beta \end{array}$$

分散

$$V\left[X\right]=\alpha {\beta }^2$$

$$\begin{array}{rl}E\left[X^2\right]& ={\int }_0^{\mathrm{\infty }}{x}^2\frac{x^{\alpha -1}}{{\beta }^{\alpha }\mathrm{\Gamma }\left(\alpha \right)}\exp (-\frac{x}{\beta })dx\\ & ={\int }_0^{\mathrm{\infty }}\frac{x^{\alpha +1}}{{\beta }^{\alpha }\mathrm{\Gamma }\left(\alpha \right)}\exp (-\frac{x}{\beta })dx\\ & ={\int }_0^{\mathrm{\infty }}\frac{(\alpha +1)\alpha {\beta }^2}{{\beta }^2}\frac{x^{\alpha +1}}{{\beta }^{\alpha }(\alpha +1)\alpha \mathrm{\Gamma }\left(\alpha \right)}\exp (-\frac{x}{\beta })dx\\ & =(\alpha +1)\alpha {\beta }^2{\int }_0^{\mathrm{\infty }}\frac{x^{\alpha }}{{\beta }^{\alpha +2}\mathrm{\Gamma }(\alpha +2)}\exp (-\frac{x}{\beta })dx\\ & =(\alpha +1)\alpha {\beta }^2\times 1\\ & =(\alpha +1)\alpha {\beta }^2\end{array}$$

より

$$V\left[X\right]=(\alpha +1)\alpha {\beta }^2-{\alpha }^2{\beta }^2=\alpha {\beta }^2$$

モーメント母関数

$$\begin{array}{rl}M\left(t\right)& =E\left[e^{tX}\right]\\ & ={\int }_0^{\mathrm{\infty }}{e}^{tx}\frac{x^{\alpha -1}}{{\beta }^{\alpha }\mathrm{\Gamma }\left(\alpha \right)}\exp (-\frac{x}{\beta })dx\\ & ={\int }_0^{\mathrm{\infty }}\frac{x^{\alpha -1}}{{\beta }^{\alpha }\mathrm{\Gamma }\left(\alpha \right)}\exp (-\frac{x}{\beta }+tx)dx\\ & ={\int }_0^{\mathrm{\infty }}\frac{x^{\alpha -1}}{{\beta }^{\alpha }\mathrm{\Gamma }\left(\alpha \right)}\exp (-\frac{x(1-\beta t)}{\beta })dx\\ & ={\int }_0^{\mathrm{\infty }}\frac{x^{\alpha -1}}{{\beta }^{\alpha }\mathrm{\Gamma }\left(\alpha \right)}\exp (-\frac{x}{\beta /(1-\beta t)})dx\\ & ={\left(\frac{1}{1-\beta t}\right)}^{\alpha }{\int }_0^{\mathrm{\infty }}\frac{x^{\alpha }}{{\left(\beta /(1-\beta t)\right)}^{\alpha }\mathrm{\Gamma }\left(\alpha \right)}\exp (-\frac{x}{\beta /(1-\beta t)})dx\\ & =\frac{1}{{(1-\beta t)}^{\alpha }}\times 1\\ & =\frac{1}{{(1-\beta t)}^{\alpha }}\end{array}$$