確率分布ごとの確率密度関数および期待値と分散

　確率分布ごとの確率密度関数 f(x), 確率変数 X の期待値 E(X) および分散 V(X) をまとめました．

超幾何分布
$\displaystyle f(x) = \frac{{}_{M}C_x\cdot{}_{N-M}C_{n-x}}{{}_{N}C_{n}}$	$\displaystyle x = Max(0, n - (N - M)), \cdots, Min(n, M)$
$\displaystyle E(X) = np$	$\displaystyle p = \frac{M}{N}$
$\displaystyle V(X) = np(1 - p)\frac{N - n}{N - 1}$
二項分布
$\displaystyle f(x) = {}_{n}C_{x}p^{x}(1 - p)^{n - x}$	$\displaystyle x = 0, 1, \cdots, n$
$\displaystyle E(X) = np$	$\displaystyle 0 < p < 1[/latex]</td> </tr> <tr> <td>[latex]\displaystyle V(X) = np(1 - p)$
ポアソン分布
$\displaystyle f(x) = \frac{\exp(- \lambda)\cdot\lambda^x}{x!}$	$\displaystyle x = 0, 1, 2, \cdots$
$\displaystyle E(X) = \lambda$	$\lambda = np$
$\displaystyle V(X) = \lambda$
幾何分布
$\displaystyle f(x) = p(1 - p)^{x - 1}$	$\displaystyle x = 1, 2, 3, \cdots$
$\displaystyle E(X) = \frac{1}{p}$	$\displaystyle 0 < p < 1[/latex]</td> </tr> <tr> <td>[latex]\displaystyle V(X) = \frac{1 - p}{p^2}$
負の二項分布
$\displaystyle f(x) = {}_{k + x - 1}C_{x}p^{k}(1-p)^{x}$	$\displaystyle x = 0, 1, 2, \cdots$
$\displaystyle E(X) = \frac{k(1 - p)}{p}$
$\displaystyle V(X) = \frac{k(1 - p)}{p^2}$
正規分布
$\displaystyle f(x) = \frac{1}{\sqrt{2\pi}\sigma}\exp\left(-\frac{(x - m)^2}{2\sigma^2}\right)$	$\displaystyle - \infty < x < \infty[/latex]</td> </tr> <tr> <td>[latex]\displaystyle E(X) = m$
$\displaystyle V(X) = \sigma^2$	$\displaystyle \sigma > 0$
指数分布
$\displaystyle f(x) = \lambda\exp(-\lambda x)$	$\displaystyle x \ge 0$
$\displaystyle f(x) = 0$	$\displaystyle x < 0[/latex]</td> </tr> <tr> <td>[latex]\displaystyle E(X) = \frac{1}{\lambda}$
$\displaystyle V(X) = \frac{1}{\lambda^2}$
ガンマ分布
$\displaystyle f(x) = \frac{\lambda^\alpha}{\Gamma(\alpha)}x^{\alpha - 1}\exp(- \lambda x)$	$\displaystyle x \ge 0$
$\displaystyle f(x) = 0$	$\displaystyle x < 0[/latex]</td> </tr> <tr> <td>[latex]\displaystyle E(X) = \frac{\alpha}{\lambda}$	$\displaystyle \lambda, \alpha > 0$
$\displaystyle V(X) = \frac{\alpha}{\lambda^2}$
ベータ分布
$\displaystyle f(x) = \frac{x^{\alpha - 1}(1 - x)^{\beta - 1}}{B(\alpha, \beta)}$	$\displaystyle 0 < x < 1[/latex]</td> </tr> <tr> <td>[latex]\displaystyle f(x) = 0$	$\displaystyle x \le 0, 1 \le x$
$\displaystyle E(X) = \frac{\alpha}{\alpha + \beta}$
$\displaystyle V(X) = \frac{\alpha\beta}{(\alpha + \beta)^2(\alpha + \beta + 1)}$
コーシー分布
$\displaystyle f(x) = \frac{\alpha}{\pi(\alpha^2 + (x - \lambda)^2)}$	$\displaystyle \alpha > 0$
対数正規分布
$\displaystyle f(x) = \frac{1}{\sqrt{2\pi}\sigma x}\exp\left(-\frac{(\log(x) - m)^2}{2\sigma^2}\right)$	$\displaystyle x > 0$
$\displaystyle f(x) = 0$	$\displaystyle x \le 0$
$\displaystyle E(X) = \exp\left(m + \frac{\sigma^2}{2}\right)$	$\displaystyle \sigma > 0$
$\displaystyle V(X) = \exp[2m + \sigma^2](\exp[\sigma^2] - 1)$
パレート分布
$\displaystyle f(x) = \frac{x_0^\alpha}{x^{\alpha + 1}}$	$\displaystyle x \ge x_0$
$\displaystyle f(x) = 0$	$\displaystyle x < 0[/latex]</td> </tr> <tr> <td>[latex]\displaystyle E(X) = \frac{ax_0}{a - 1}$	$\displaystyle a > 1$
$\displaystyle V(X) = \frac{ax_0^2}{a - 2} - \left(\frac{ax_0}{a - 1}\right)^2$	$\displaystyle a > 2$
ワイブル分布
$\displaystyle f(x) = \frac{\gamma x^{\gamma - 1}}{\lambda^b}\exp\left[-\left(\frac{x}{y}\right)^\gamma\right]$	$\displaystyle x \ge 0$
$\displaystyle f(x) = 0$	$\displaystyle x < 0[/latex]</td> </tr> <tr> <td>[latex]\displaystyle E(X) = \lambda\Gamma\left(1 + \frac{1}{\gamma}\right)$	$\displaystyle \lambda > 0$
$\displaystyle V(X) = \lambda^2\left[\Gamma\left(2 + \frac{1}{\gamma}\right) - \left(\Gamma\left(1 + \frac{1}{\gamma}\right)\right)^2\right]$	$\displaystyle \gamma > 0$