方差

定义

$$D(X)=E(X-\mu_X)^2$$

期望

$$D(X)=E(X^2)-(EX)^2$$

证明：

$$ \begin{aligned} D(X)&=E(X-\mu_X)^2 \\ &=E(X^2-2X\mu_X+\mu_X^2) \\ &=E(X^2)-2\mu_XE(X)+\mu_X^2 \\ &=E(X^2)-(E(X))^2 \end{aligned} $$

随机变量和的方差

$$D(X+Y)=D(X)+D(Y)+2Cov(X,Y)$$

证明：

$$\begin{aligned} D(X+Y)&=E(X+Y-(\mu_X+\mu_Y))^2 \\ &=E(X^2+2XY+Y^2-\mu_X^2-\mu_Y^2-2\mu_X\mu_Y) \\ &=E(X^2)-\mu_X^2+E(Y^2)-\mu_Y^2+2E(XY)-2\mu_X\mu_Y\\ &=D(X)+D(Y)-2Cov(X,Y) \end{aligned}$$

采样均值的方差

$$\overline{X} =\frac{1}{n}\sum_i X_i$$

$$\begin{aligned} D(\overline{X}) &= D(\frac{1}{n}\sum_iX_i) \\ &= \sum_i\frac{1}{n^2}D(X) \\ &= \frac{1}{n}D(X) \end{aligned}$$

协方差

定义

$$ Cov(X,Y) = E((X-\mu_X)(Y-\mu_Y)) $$

期望

$$Cov(X,Y) = E(XY)-E(X)E(Y)$$

证明：

$$ \begin{aligned} Cov(X,Y)&=E(XY)-\mu_YE(X)-\mu_XE(Y)+\mu_X\mu_Y \\ &=E(XY)-E(X)E(Y) \end{aligned} $$

独立随机变量

两个独立随机变量的协方差为0

方差的无偏估计

$\frac{\sum_i (x_i-\overline{x})^2}{n-1}$是$\mathbb{D}X$的无偏估计

证明：

$$ \begin{aligned} E(\sum_i(x_i-\overline{x})^2) &= E(\sum_i(X_i^2)-2\overline{x}\sum_i(X_i)+n\overline{x}^2) \\ &= E(\sum_i(X_i^2)-2n\overline{x}^2+n\overline{x}^2) \\ &= nE(X^2)-nE(\overline{X}^2) \\ &= n(E(X^2)-(EX)^2)-n(E\overline{X}^2-EX^2) \\ &= nD(X)-D(X) \\ &= (n-1)D(X) \end{aligned} $$