4.3 常见分布的期望与方差

分布	分布律	期望	方差
$0-1$ 分布 $B(1,p)$	$P(X=k)=p^k(1-p{)}^{1-k}$ $k=0,1,0<p<1$	$p$	$p(1-p)$
二项分布 $B(n,p)$	$P(X=k)={\mathrm{C}}_n^k{p}^k(1-p{)}^{n-k}$ $k=0,1,\cdots ,n,0<p<1$	$np$	$np(1-p)$
泊松分布 $P(\lambda )$	$P(X=k)={\mathrm{e}}^{-\lambda }{\displaystyle \frac{{\lambda }^k}{k!}}$ $k=0,1,2,\cdots ,\lambda >0$	$\lambda$	$\lambda$
⼏何分布 $G(p)$	$P(X=k)=(1-p{)}^{k-1}p$ $k=1,2,\cdots ,0<p<1$	${\displaystyle \frac{1}{p}}$	${\displaystyle \frac{1}{p^2}}-{\displaystyle \frac{1}{p}}$
超⼏何分布 $H(n,M,N)$
负二项分布 (Pascal 分布)	$P(X=k)={\mathrm{C}}_{k-1}^{r-1}{p}^r(1-p{)}^{k-r}$ $k=r,r+1,\cdots ,0<p<1$	${\displaystyle \frac{r}{p}}$	$r({\displaystyle \frac{1}{p^2}}-{\displaystyle \frac{1}{p}})$
均匀分布 $U(a,b)$	${\displaystyle f(x)=\{\begin{array}{ll}{\displaystyle \frac{1}{b-a}},& a<x<b,\\ 0,& \text{otherwise}.\end{array}}$	${\displaystyle \frac{a+b}{2}}$	${\displaystyle \frac{(b-a{)}^2}{12}}$
指数分布 $E(\lambda )$	$f(x)=\{\begin{array}{ll}\lambda {\mathrm{e}}^{-\lambda x},& x>0,\\ 0,& \text{otherwise}.\end{array}$	${\displaystyle \frac{1}{\lambda }}$	${\displaystyle \frac{1}{{\lambda }^2}}$
正态分布 $N(\mu ,{\sigma }^2)$	$f(x)={\displaystyle \frac{1}{\sqrt{2\pi }\sigma }}\exp (-{\displaystyle \frac{(x-\mu {)}^2}{2{\sigma }^2}})$ $-\mathrm{\infty }<x<+\mathrm{\infty },-\mathrm{\infty }<\mu <\mathrm{\infty },\sigma >0$	$\mu$	${\sigma }^2$