正态总体统计量的分布特征

$X_{1}, X_{2}, \dots, X_{n}$ 是取值正态总体 $X \sim N (μ, σ^{2})$ 的样本
$E (X_{i}) = μ$ $D (X_{i}) = σ$

\begin{array}{r} \frac{X_{i} - μ}{σ} \sim N (0, 1) \sum_{i = 1}^{n} \frac{X_{i} - μ}{σ} \sim N (0, n) \end{array}

样本均值 $\overset{―}{X}$ 与样本方差 $S^{2}$ 相互独立

一、单正态总体

1. 样本均值的分布（已知总体的方差）

样本均值 $\overset{―}{X}$ 满足：
$E (\overset{―}{X}) = μ$ $D (\overset{―}{X}) = \frac{σ^{2}}{n}$
服从正态分布

\begin{array}{r} \overset{―}{X} \sim N (μ, \frac{σ^{2}}{n}) \frac{\overset{―}{X} - μ}{\frac{σ}{\sqrt{n}}} \sim N (0, 1) \end{array}

2. 样本方差的分布

服从卡方分布

\begin{array}{r} (n - 1) S^{2} = \sum_{i = 1}^{n} (X_{i} - \overset{―}{X})^{2} \frac{(n - 1) S^{2}}{σ^{2}} = \sum_{i = 1}^{n} (\frac{X_{i} - \overset{―}{X}}{σ})^{2} \sim χ^{2} (n - 1) \end{array}

3. 样本均值的分布（未知总体的方差）

服从 t分布

\begin{aligned} \frac{\overset{―}{X} - μ}{\frac{S}{\sqrt{n}}} & = \frac{\overset{―}{X} - μ}{\frac{σ}{\sqrt{n}}} \cdot \frac{1}{\sqrt{\frac{(n - 1) S^{2}}{σ^{2} (n - 1)}}} = \frac{\sim N (0, 1)}{\sqrt{\frac{\sim χ^{2} (n - 1)}{n - 1}}} \sim t (n - 1) \end{aligned}

二、双正态总体

$X_{1}, X_{2}, \dots, X_{n}$ 是取值正态总体 $X \sim N (μ_{1}, σ_{1}^{2})$ 的样本，样本均值 $\overset{―}{X}$ ，样本方差 $S_{X}^{2}$
$Y_{1}, Y_{2}, \dots, Y_{n}$ 是取值正态总体 $Y \sim N (μ_{2}, σ_{2}^{2})$ 的样本，样本均值 $\overset{―}{Y}$ ，样本方差 $S_{Y}^{2}$

1. 样本均值差的分布

如果 $σ_{1}^{2} = σ_{2}^{2}$ 两个正态总体的方差相等

\begin{aligned} \overset{―}{X} \sim N (μ_{1}, \frac{σ^{2}}{n_{1}}) \overset{―}{Y} \sim N (μ_{2}, \frac{σ^{2}}{n_{2}}) \\ \overset{―}{X} - \overset{―}{Y} \sim N (μ_{1} - μ_{2}, \frac{σ^{2}}{n_{1}} + \frac{σ^{2}}{n_{2}}) \\ \frac{\overset{―}{X} - \overset{―}{Y} - (μ_{1} - μ_{2})}{σ \sqrt{\frac{1}{n_{1}} + \frac{1}{n_{2}}}} \sim N (0, 1) \\ \frac{(n_{1} - 1) S_{X}^{2}}{σ^{2}} + \frac{(n_{2} - 1) S_{Y}^{2}}{σ^{2}} \\ \sim χ^{2} (n_{1} + n_{2} - 2) \end{aligned}

\begin{aligned} \frac{\overset{―}{X} - \overset{―}{Y} - (μ_{1} - μ_{2})}{\sqrt{\frac{(n_{1} - 1) S_{X}^{2} + (n_{2} - 1) S_{Y}^{2}}{n_{1} + n_{2} - 2}} \sqrt{\frac{1}{n_{1}} + \frac{1}{n_{2}}}} & = \frac{\sim N (0, 1)}{\sqrt{\frac{\sim χ^{2} (n_{1} + n_{2} - 2)}{n_{1} + n_{2} - 2}}} \sim t (n_{1} + n_{2} - 2) \end{aligned}

服从 t分布

2. 样本方差之比的分布

\begin{array}{r} \frac{S_{X}^{2} / σ_{1}^{2}}{S_{Y}^{2} / σ_{2}^{2}} = \frac{\frac{(n_{1} - 1) S_{X}^{2}}{σ_{1}^{2} (n_{1} - 1)}}{\frac{(n_{2} - 1) S_{Y}^{2}}{σ_{2}^{2} (n_{2} - 1)}} = \frac{\frac{\sim χ^{2} (n_{1} - 1)}{(n_{1} - 1)}}{\frac{\sim χ^{2} (n_{2} - 1)}{(n_{2} - 1)}} \sim F (n_{1} - 1, n_{2} - 1) \end{array}

服从 F分布