数一概率论 八大抽样分布

八大抽样分布

单正态分布

$μ$估计

$σ^2$已知

$$\frac {\overline X - μ}{σ/\sqrt n}\sim N(0,1)$$

$σ^2$未知

$$\frac {\overline X - μ}{S/\sqrt n}\sim t_{n-1}$$

$σ^2$估计

$μ$已知

$$\frac {\sum_{i=1}^{n}(X_i - μ)^2}{σ^2}\sim χ^2_{n}$$

$μ$未知

$$\frac {\sum_{i=1}^{n}(X_i- \overline X)^2}{σ^2}=\frac {(n-1)S^2}{σ^2}\sim χ^2_{n-1}$$

二正态总体

$μ_1-μ_2$估计

$σ_1$,$σ_2$已知

$$\frac {(\overline X-\overline Y)-(μ_1-μ_2)}{\sqrt{\frac{σ_1^2}{n}+\frac{σ_2^2}{m}}}\sim N(0,1)$$

$σ_1$,$σ_2$未知，且都为$σ$

$$\frac {(\overline X-\overline Y)-(μ_1-μ_2)}{\sqrt{\frac{1}{n}+\frac{1}{m}} \sqrt{\frac{(n-1)S_1^2+(m-1)S_2^2}{m+n-2}}}\sim t_{m+n-2}$$

$\frac {σ_1^2}{σ_2^2}$估计

$μ_1$,$μ_2$已知

$$\frac{\frac{\sum_{i=1}^n(X_i-μ_1)^2}{nσ_1^2}}{\frac{\sum_{i=1}^m(Y_i-μ_2)^2}{mσ_2^2}}\sim F_{n,m}$$

$μ_1$,$μ_2$未知

$$\frac {S_2^2/σ_2^2}{S_1^2/σ_1^2}\sim F_{m-1,n-1}$$