常见概率分布及其数学期望和方差

以下是常见概率分布及其期望和方差公式的表格：

分布名称	分布列或概率密度	期望	方差
离散型分布
0-1分布（两点分布或伯努利分布）\(B(1, p)\)	\(p_{k}=p^{k}(1-p)^{1-k},k = 0,1\)	\(p\)	\(p(1-p)\)
二项分布\(B(n,p)\)	\(p_{k}=\binom{n}{k}p^{k}(1-p)^{n-k},k = 0,1,\cdots,n\)	\(np\)	\(np(1-p)\)
泊松分布\(P(\lambda)\)	\(p_{k}=\frac{\lambda^{k}}{k!}e^{-\lambda},k = 0,1,\cdots\)	\(\lambda\)	\(\lambda\)
[[超几何分布]]\(H(n,N,M)\)	\(p_{k}=\frac{\binom{M}{k}\binom{N-M}{n-k}}{\binom{N}{n}},k = 0,1,\cdots,r,r=\min{(M,n)}\)	\(\frac{nM}{N}\)	\(\frac{nM(N-M)(N-n)}{N^{2}(N-1)}\)
[[几何分布]]\(Ge(p)\)	\(p_{k}=(1-p)^{k-1}p,k = 1,2,\cdots\)	\(\frac{1}{p}\)	\(\frac{1-p}{p^{2}}\)
负二项分布\(Nb(r,p)\)	\(p_{k}=\binom{k-1}{r-1}(1-p)^{k-r}p^{r},k = r,r + 1,\cdots\)	\(\frac{r}{p}\)	\(\frac{r(1-p)}{p^{2}}\)
连续型分布
均匀分布\(U(a,b)\)	\(f(x)=\frac{1}{b-a},a<x<b\)	\(\frac{a + b}{2}\)	\(\frac{(b-a)^{2}}{12}\)
正态分布\(N(\mu,\sigma^{2})\)	\(f(x)=\frac{1}{\sqrt{2\pi}\sigma}\exp\left\{-\frac{(x-\mu)^{2}}{2\sigma^{2}}\right\},-\infty<x<+\infty\)	\(\mu\)	\(\sigma^{2}\)
指数分布\(Exp(\lambda)\)	\(f(x)=\lambda e^{-\lambda x},x>0\)	\(\frac{1}{\lambda}\)	\(\frac{1}{\lambda^{2}}\)
伽马分布\(\Gamma(\alpha,\lambda)\)	\(f(x)=\frac{\lambda^{\alpha}}{\Gamma(\alpha)}x^{\alpha-1}e^{-\lambda x},x>0\)	\(\frac{\alpha}{\lambda}\)	\(\frac{\alpha}{\lambda^{2}}\)
卡方分布\(\chi^{2}(n)\)	\(f(x)=\frac{1}{2^{\frac{n}{2}}\Gamma(\frac{n}{2})}x^{\frac{n}{2}-1}e^{-\frac{x}{2}},x>0\)	\(n\)	\((2n)\)
t分布\(t(n)\)	\(f(x)=\frac{\Gamma(\frac{n + 1}{2})}{\sqrt{n\pi}\Gamma(\frac{n}{2})}(1+\frac{x^{2}}{n})^{-\frac{n + 1}{2}}\)	\(0（n>1时）\)	\(\frac{n}{n - 2}（n>2时）\)
F分布\(F(m,n)\)	\(f(x)=\frac{\Gamma(\frac{m + n}{2})}{\Gamma(\frac{m}{2})\Gamma(\frac{n}{2})}(\frac{m}{n})^{\frac{m}{2}}x^{\frac{m}{2}-1}(1+\frac{mx}{n})^{-\frac{m + n}{2}},x>0\)	\(\frac{n}{n - 2}（n>2时）\)	\(\frac{2n^{2}(m + n - 2)}{m(n - 2)^{2}(n - 4)}（n>4时）\)