# ガンマ分布
###### tags: `probability-theory`
## 確率密度関数

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

## 確率密度関数の形状

ガンマ分布の密度関数は、$x\rightarrow\infty$で$0$に収束する。
$$
\lim_{x\rightarrow\infty} f\left(x\right) = 0
$$
また$x\rightarrow 0$の挙動は$\alpha$の$1$との大小関係で定まる。
$$
\lim_{x\rightarrow 0} f\left(x\right) = \left\{
\begin{array}{ll}
0 & \alpha>1 \\
\infty & \alpha<1
\end{array}
\right.
$$

密度関数の導関数
$$
\begin{align}
\frac{\partial}{\partial x} f\left(x\right)
&= \frac{\left(\alpha-1\right)x^{\alpha-2}}{\beta^\alpha \Gamma\left(\alpha\right)} \exp\left(-\frac{x}{\beta}\right) - \frac{x^{\alpha-1}}{\beta^{\alpha+1} \Gamma\left(\alpha\right)} \exp\left(-\frac{x}{\beta}\right) \notag \\
&= \frac{x^{\alpha-2}}{\beta^\alpha \Gamma\left(\alpha\right)}\exp\left(-\frac{x}{\beta}\right)\times \left(\alpha-1-\frac{x}{\beta} \right)
\end{align}
$$
は$\alpha\leq 1$ならば常に負、$\alpha>1$ならば$x=\left(\alpha-1\right)\beta$で符号が正から負に変わる。

## 平均

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

$$
\begin{align}
E\left[X\right]
&= \int_{0}^{\infty} x \frac{x^{\alpha-1}}{\beta^\alpha \Gamma\left(\alpha\right)} \exp\left(-\frac{x}{\beta}\right)dx \notag \\
&= \int_{0}^{\infty} \frac{x^{\alpha}}{\beta^\alpha \Gamma\left(\alpha\right)} \exp\left(-\frac{x}{\beta}\right)dx \notag \\
&= \int_{0}^{\infty} \frac{\alpha\beta}{\beta} \frac{x^{\alpha}}{\beta^\alpha \alpha \Gamma\left(\alpha\right)} \exp\left(-\frac{x}{\beta}\right)dx \notag \\
&= \alpha\beta \int_{0}^{\infty} \frac{x^{\alpha}}{\beta^{\alpha+1} \Gamma\left(\alpha+1\right)} \exp\left(-\frac{x}{\beta}\right)dx \notag \\
&= \alpha\beta \times 1 \notag \\
&= \alpha\beta
\end{align}
$$

## 分散

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

$$
\begin{align}
E\left[X^2\right]
&= \int_{0}^{\infty} x^2 \frac{x^{\alpha-1}}{\beta^\alpha \Gamma\left(\alpha\right)} \exp\left(-\frac{x}{\beta}\right)dx \notag \\
&= \int_{0}^{\infty} \frac{x^{\alpha+1}}{\beta^\alpha \Gamma\left(\alpha\right)} \exp\left(-\frac{x}{\beta}\right)dx \notag \\
&= \int_{0}^{\infty} \frac{\left(\alpha+1\right)\alpha\beta^2}{\beta^2} \frac{x^{\alpha+1}}{\beta^\alpha \left(\alpha+1\right) \alpha \Gamma\left(\alpha\right)} \exp\left(-\frac{x}{\beta}\right)dx \notag \\
&= \left(\alpha+1\right)\alpha\beta^2 \int_{0}^{\infty} \frac{x^{\alpha}}{\beta^{\alpha+2} \Gamma\left(\alpha+2\right)} \exp\left(-\frac{x}{\beta}\right)dx \notag \\
&= \left(\alpha+1\right)\alpha\beta^2 \times 1 \notag \\
&= \left(\alpha+1\right)\alpha\beta^2
\end{align}
$$

より

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

## モーメント母関数

$$
\begin{align}
M\left(t\right) &= E\left[e^{tX}\right] \notag \\
&= \int_{0}^{\infty} e^{tx} \frac{x^{\alpha-1}}{\beta^\alpha \Gamma\left(\alpha\right)} \exp\left(-\frac{x}{\beta}\right)dx \notag \\
&= \int_{0}^{\infty} \frac{x^{\alpha-1}}{\beta^\alpha \Gamma\left(\alpha\right)} \exp\left(-\frac{x}{\beta}+tx\right)dx \notag \\
&= \int_{0}^{\infty} \frac{x^{\alpha-1}}{\beta^\alpha \Gamma\left(\alpha\right)} \exp\left(-\frac{x\left(1-\beta t\right)}{\beta}\right)dx \notag \\
&= \int_{0}^{\infty} \frac{x^{\alpha-1}}{\beta^\alpha \Gamma\left(\alpha\right)} \exp\left(-\frac{x}{\beta/\left(1-\beta t\right)}\right)dx \notag \\
&= \left(\frac{1}{1-\beta t}\right)^\alpha \int_{0}^{\infty} \frac{x^{\alpha}}{\left(\beta/\left(1-\beta t\right)\right)^\alpha \Gamma\left(\alpha\right)} \exp\left(-\frac{x}{\beta/\left(1-\beta t\right)}\right)dx \notag \\
&= \frac{1}{\left(1-\beta t\right)^\alpha} \times 1 \notag \\
&= \frac{1}{\left(1-\beta t\right)^\alpha}
\end{align}
$$