常用分布的期望、方差和特征函数
概率分布表（期望与方差）
| 分 布 | 分布列 \(p_k\) 或分布密度 \(p(x)\) | 期 望 | 方 差 |
|---|---|---|---|
| 0－1 分布 | \(p_k=p^k(1-p)^{1-k}, \quad k=0,1\) | \(p\) | \(p(1-p)\) |
| 二项分布 \(b(n, p)\) |
\(P_k=\displaystyle\binom{n}{k} p^k(1-p)^{n-k}, \quad k=0,1, \cdots, n\) | \(n p\) | \(n p(1-p)\) |
| 泊松分布 \(P(\lambda)\) |
\(p_k=\dfrac{\lambda^k}{k!} \mathrm{e}^{-\lambda}, \quad k=0,1, \cdots\) | \(\lambda\) | \(\lambda\) |
| 超几何分布 \(h(n, N, M)\) |
\(p_k=\dfrac{\displaystyle\binom{M}{k}\displaystyle\binom{N-M}{n-k}}{\displaystyle\binom{N}{n}}, \quad \begin{aligned} k=0,1, \cdots, r, \\ r=\min \{M, n\} \end{aligned}\) | \(n \dfrac{M}{N}\) | \(\dfrac{n M(N-M)(N-n)}{N^2(N-1)}\) |
| 几何分布 \(Ge(p)\) |
\(p_k=(1-p)^{k-1} p, \quad k=1,2, \cdots\) | \(\dfrac{1}{p}\) | \(\dfrac{1-p}{p^2}\) |
| 负二项分布 \(Nb(r, p)\) |
\(p_k=\displaystyle\binom{k-1}{r-1}(1-p)^{k-1} p^{r}, \quad k=r, r+1, \cdots\) | \(\dfrac{r}{p}\) | \(\dfrac{r(1-p)}{p^2}\) |
| 正态分布 \(N(\mu, \sigma^2)\) |
\(p(x)=\dfrac{1}{\sqrt{2\pi}\sigma}\exp\left\{-\dfrac{(x-\mu)^2}{2\sigma^2}\right\}\) | \(\mu\) | \(\sigma^2\) |
| 均匀分布 \(U(a, b)\) |
\(p(x)=\dfrac{1}{b-a}, \quad a<x<b\) | \(\dfrac{a+b}{2}\) | \(\dfrac{(b-a)^2}{12}\) |
| 指数分布 \(\operatorname{Exp}(\lambda)\) |
\(p(x)=\lambda \mathrm{e}^{-\lambda x}, \quad x \geqslant 0\) | \(\dfrac{1}{\lambda}\) | \(\dfrac{1}{\lambda^2}\) |
| 伽马分布 \(Ga(\alpha, \lambda)\) |
\(p(x)=\dfrac{\lambda^\alpha}{\Gamma(\alpha)} x^{\alpha-1} \mathrm{e}^{-\lambda x}, \quad x \geqslant 0\) | \(\dfrac{\alpha}{\lambda}\) | \(\dfrac{\alpha}{\lambda^2}\) |
| \(\chi^2(n)\) 分布 | \(p(x)=\dfrac{x^{n / 2-1} \mathrm{e}^{-x / 2}}{\Gamma(n / 2) 2^{n / 2}}, \quad x \geqslant 0\) | \(n\) | \(2n\) |
| 贝塔分布 \(Be(a, b)\) |
\(p(x)=\dfrac{\Gamma(a+b)}{\Gamma(a) \Gamma(b)} x^{a-1}(1-x)^{b-1}, \quad 0<x<1\) | \(\dfrac{a}{a+b}\) | \(\dfrac{ab}{(a+b)^2(a+b+1)}\) |
| 对数正态分布 \(LN(\mu, \sigma^2)\) |
\(p(x)=\dfrac{1}{\sqrt{2 \pi} \sigma x} \exp \left\{-\dfrac{(\ln x-\mu)^2}{2 \sigma^2}\right\}, x>0\) | \(\mathrm{e}^{\mu+\sigma^2/2}\) | \(\mathrm{e}^{2\mu+\sigma^2}\left(\mathrm{e}^{\sigma^2}-1\right)\) |
| 柯西分布 \(\operatorname{Cau}(\mu, \lambda)\) |
\(p(x)=\dfrac{1}{\pi} \dfrac{\lambda}{\lambda^2+(x-\mu)^2}, -\infty<x<\infty\) | 不存在 | 不存在 |
| 韦布尔分布 | \(\begin{aligned} p(x)&=F'(x), \\ F(x)&=1-\exp\left\{-\left(\dfrac{x}{\eta}\right)^m\right\}, x>0 \end{aligned}\) | \(\eta \Gamma\left(1+\dfrac{1}{m}\right)\) | \(\eta^2\left[\Gamma\left(1+\dfrac{2}{m}\right)-\Gamma^2\left(1+\dfrac{1}{m}\right)\right]\) |
概率分布表（特征函数）
| 分 布 | 分布列 \(p_k\) 或分布密度 \(p(x)\) | 特征函数 \(\varphi(t)\) |
|---|---|---|
| 单点分布 | \(P(X=a)=1\) | \(\mathrm{e}^{\text{i} ta}\) |
| 0－1 分布 | \(p_k=p^k q^{1-k}, q=1-p, k=0,1\) | \(p \mathrm{e}^{\text{i} t}+q\) |
| 二项分布 \(b(n, p)\) |
\(p_k=\displaystyle\binom{n}{k} p^k q^{n-k}, \quad k=0,1, \cdots, n\) | \(\left(p \mathrm{e}^{\text{i} t}+q\right)^n\) |
| 泊松分布 \(P(\lambda)\) |
\(p_k=\dfrac{\lambda^k}{k!} \mathrm{e}^{-\lambda}, \quad k=0,1, \cdots\) | \(\mathrm{e}^{\lambda\left(\mathrm{e}^{\text{i} t}-1\right)}\) |
| 几何分布 \(Ge(p)\) |
\(p_k=p q^{k-1}, k=1,2, \cdots\) | \(p /\left(1-q \mathrm{e}^{\text{i} t}\right)\) |
| 负二项分布 \(Nb(r, p)\) |
\(p_k=\displaystyle\binom{k-1}{r-1} p^{r} q^{k-r}, \quad k=r, r+1, \cdots\) | \(\left(\dfrac{p}{1-q \mathrm{e}^{\text{i} t}}\right)^{r}\) |
| 均匀分布 \(U(a, b)\) |
\(p(x)=\dfrac{1}{b-a}, \quad a<x<b\) | \(\dfrac{\mathrm{e}^{\text{i}bt}-\mathrm{e}^{\text{i}at}}{\text{i}t(b-a)}\) |
| 正态分布 \(N(\mu, \sigma^2)\) |
\(p(x)=\dfrac{1}{\sqrt{2 \pi} \sigma} \exp \left\{-\dfrac{(x-\mu)^2}{2 \sigma^2}\right\}\) | \(\exp \left(\text{i} \mu t-\dfrac{\sigma^2 t^2}{2}\right)\) |
| 指数分布 \(\operatorname{Exp}(\lambda)\) |
\(p(x)=\lambda \mathrm{e}^{-\lambda x}, \quad x \geqslant 0\) | \(\left(1-\dfrac{\text{i}t}{\lambda}\right)^{-1}\) |
| 伽马分布 \(Ga(\alpha, \lambda)\) |
\(p(x)=\dfrac{\lambda^\alpha}{\Gamma(\alpha)} x^{\alpha-1} \mathrm{e}^{-\lambda x}, \quad x \geqslant 0\) | \(\left(1-\dfrac{\text{i} t}{\lambda}\right)^{-\alpha}\) |
| \(\chi^2(n)\) 分布 | \(p(x)=\dfrac{x^{n/2-1} \mathrm{e}^{-x/2}}{\Gamma(n/2) 2^{n/2}}, \quad x \geqslant 0\) | \((1-2 \text{i} t)^{-n/2}\) |
| 贝塔分布 \(Be(a, b)\) |
\(p(x)=\dfrac{\Gamma(a+b)}{\Gamma(a) \Gamma(b)} x^{a-1}(1-x)^{b-1}, \quad 0<x<1\) | \(\dfrac{\Gamma(a+b)}{\Gamma(a)} \sum\limits_{k=0}^{\infty} \dfrac{(\text{i}t)^k \Gamma(a+k)}{k!\Gamma(a+b+k) \Gamma(k+1)}\) |
| 柯西分布 \(\operatorname{Cau}(0,1)\) |
\(p(x)=\dfrac{1}{\pi(1+x^2)}, \quad -\infty<x<\infty\) | \(\mathrm{e}^{-\|t\|}\) |
评论