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\(F(m,n)\) | \(f(x)=\frac{\Gamma(\frac{m + n}{2})}{\Gamma(\frac{m}{2})\Gamma(\frac{n}{2})}(\frac{m}{n})^{\frac{m}{2}}x^{\frac{m}{2}-1}(1+\frac{mx}{n})^{-\frac{m + n}{2}},x>0\) | \(\frac{n}{n - 2}（n>2时）\) | \(\frac{2n^{2}(m + n - 2)}{m(n - 2)^{2}(n - 4)}（n>4时）\) |