常用分布

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\begin{tabular}{cccc}

    分 布 & 分布列 $p_k$ 或分布密度 $p(x)$ & 期 望 & 方 差 \\

    0－1 分布 & $p_k=p^k(1-p)^{1-k}, \quad k=0,1$ & $p$ & $p(1-p)$ \\
    $
    \begin{gathered}
        \text { 二项分布 } \\
        b(n, p)
    \end{gathered}
    $ & $P_k=\displaystyle{\binom{n}{k}} p^k(1-p)^{n-k}, \quad k=0,1, \cdots, n$ & $n p$ & $n p(1-p)$ \\
    $
    \begin{gathered}
        \text { 泊松分布 } \\
        P(\lambda)
    \end{gathered}
    $ & $p_k=\dfrac{\lambda^k}{k!} \mathrm{e}^{-\lambda}, \quad k=0,1, \cdots$ & $\lambda$ & $\lambda$ \\
    $
    \begin{gathered}
        \text { 超几何分布 } \\
        h(n, N, M)
    \end{gathered}
    $& $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}
    $ & $n \dfrac{M}{N}$ & $\dfrac{n M(N-M)(N-n)}{N^2(N-1)}$ \\
    $
    \begin{gathered}
        \text { 几何分布 } \\
        G e(p)
    \end{gathered}
    $ & $p_k=(1-p)^{k-1} p, \quad k=1,2, \cdots$ & $\dfrac{1}{p}$ & $\dfrac{1-p}{p^2}$ \\
     $
     \begin{gathered}
        \text { 负二项分布 } \\mathbb{N} b(r, p)
     \end{gathered}
     $

     & $p_k=\displaystyle\binom{k-1}{r-1}(1-p)^{k-1} p^{\prime}, \quad k=r, r+1, \cdots$ & $\dfrac{r}{p}$ & $\dfrac{r(1-p)}{p^2}$ \\
    $
    \begin{gathered}
        \text { 正态分布 } \\
        N\left(\mu, \sigma^2\right)
    \end{gathered}
    $& $p(x)=\dfrac{1}{\sqrt{2\pi}\sigma}\exp\left\lbrace-\dfrac{(x-\mu)^2}{2\sigma^2}\right\rbrace$ & $\mu$ & $\sigma^2$ \\
    $
    \begin{gathered}
        \text { 均匀分布 } \\
        U(a, b)
    \end{gathered}
    $ & $p(x)=\dfrac{1}{b-a}, \quad a<x<b$ & $\dfrac{a+b}{2}$ & $\dfrac{(b-a)^2}{12}$ \\

    $
    \begin{gathered}
        \text { 指数分布 } \\
        \operatorname{Exp}(\lambda)
    \end{gathered}
    $
    & $p(x)=\lambda \mathrm{e}^{-\lambda x}, \quad x \geqslant 0$ & $\dfrac{1}{\lambda}$ & $\dfrac{1}{\lambda^2}$ \\
    $
    \begin{gathered}
        \text { 伽马分布 } \\
        Ga(\alpha, \lambda)
    \end{gathered}
    $
    & $p(x)=\dfrac{\lambda^\alpha}{\Gamma(\alpha)} x^{\alpha-1} \mathrm{e}^{-\lambda x}, \quad x \geqslant 0$ & $\dfrac{\alpha}{\lambda}$ & $\dfrac{\alpha}{\lambda^2}$ \\
    $\Chi^2(n)$ 分布 & $p(x)=\dfrac{x^{n / 2-1} \mathrm{e}^{-x / 2}}{\Gamma(n / 2) 2^{n / 2}}, \quad x \geqslant 0$ & $n$ & $2 n$ \\
    $
    \begin{gathered}
        \text { 贝塔分布 } \\
        Be(a, b)
    \end{gathered}
    $
    & $p(x)=\dfrac{\Gamma(a+b)}{\Gamma(a) \Gamma(b)} x^{a-1}(1-x)^{b-1}, \quad 0<x<1$ & $\dfrac{a}{a+b}$ & $\dfrac{a b}{(a+b)^2(a+b+1)}$ \\
    $
    \begin{gathered}
        \text { 对数正态分布 } \\
        LN\left(\mu, \sigma^2\right)
    \end{gathered}
    $
    & $p(x)=\dfrac{1}{\sqrt{2 \pi} \sigma x} \exp \left\{-\dfrac{(\ln x-\mu)^2}{2 \sigma^2}\right\}, x>0$ & $\mathrm{e}^{\mu+\sigma^2/2}$ & $\mathrm{e}^{2\mu+\sigma^2}\left(\mathrm{e}^{\sigma^2}-1\right)$ \\
    $
    \begin{gathered}
        \text { 柯西分布 } \\
        \operatorname{Cau}(\mu, \lambda)
    \end{gathered}
    $
     & $p(x)=\dfrac{1}{\pi} \dfrac{\lambda}{\lambda^2+(x-\mu)^2},-\infty<x<\infty$ & 不存在 & 不存在 \\
     韦布尔分布 & $
     \begin{gathered}
        p(x)=F^{\prime}(x),\\
         F(x)=1-\exp \left\{-\left(\dfrac{x}{\eta}\right)^m\right\}, x>0
     \end{gathered}$ & $\eta \Gamma\left(1+\dfrac{1}{m}\right)$ & $
    \begin{gathered}
        \eta^2\left[\Gamma\left(1+\dfrac{2}{m}\right)-\right. \\
        \left.\Gamma^2\left(1+\dfrac{1}{m}\right)\right]
    \end{gathered}
    $ \\

\end{tabular}
}

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        \begin{tabular}{ccc}

             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 & $p_k=p^k q^{1-k}, q=1-p, k=0,1$. & $p \mathrm{e}^{\text{i} t}+q$ \\
             $
            \begin{gathered}
                \text { 二项分布 } \\
                b(n, p)
            \end{gathered}
            $ & $p_k=\displaystyle\binom{n}{k} p^k q^{n-k}, \quad k=0,1, \cdots, n$. & $\left(p \mathrm{e}^{\text{i} t}+q\right)^n$ \\
             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 $G e(p)$ & $p_k=p q^{k-1}, k=1,2, \cdots$. & $p /\left(1-q \mathrm{e}^{\text { i}t}\right)$  \\
             负二项分布 $N b(r, p)$ & $p_k=\binom{k-1}{r-1} p^{\prime} q^{k-r}, \quad k=r, r+1, \cdots$. & $\left(\dfrac{p}{1-q e^{\text{i} t}}\right)^{\prime}$ \\
             均匀分布
            $
            U(a, b)
            $ & $p(x)=\dfrac{1}{b-a}, \quad a<x<b$. & $\dfrac{\mathrm{e}^{\mathrm{i}bt}-\mathrm{e}^{\mathrm{i}at}}{\mathrm{i}t(b-a)}$ \\
             正态分布
            $
            N\left(\mu, \sigma^2\right)
            $ & $
            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)
            $ & $p(x)=\lambda \mathrm{e}^{-\lambda x}, \quad x \geqslant 0$. & $\left(1-\dfrac{\mathrm{i}t}{\lambda}\right)^{-1}$ \\
             伽马分布
            $
            G a(\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
            $
            B e(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{(\mathrm{i}t)^k \Gamma(a+k)}{k!\Gamma(a+b+k) \Gamma(k+1)}
            $ \\
             柯西分布
            $
            \operatorname{Cau}(0,1)
            $ & $p(x)=\dfrac{1}{\pi\left(1+x^2\right)}, \quad-\infty<x<\infty$ & $e^{-|t|}$ \\

        \end{tabular}
    }