Section04_协方差

基本概念

定义

$$\mathrm{Cov}(X,Y) = E[(X-EX)(Y-EY)] \\$$

• 特别地 $\mathrm{Cov}(X,X) = E[(X-EX)(X-EX)] = DX$

计算

$$\mathrm{Cov}(X,Y) = EXY - EX\cdot EY = \rho_{XY}\cdot \sqrt{DX}\sqrt{DY}$$

• 注 $EXY = \begin{cases} \displaystyle \sum_{i}^{}\sum_{j}^{}x_{i}y_{j}\mathbb{P}\{X=x_{i}, Y = y_{j}\} \\ \displaystyle \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}xyf(x,y)\cdot dxdy \end{cases}$

性质

1. $\mathrm{Cov}(X,c) = 0; \quad c\text{为常数}$
   • 特别地 $\mathrm{Cov}(X, EY) = 0$
2. $\mathrm{Cov}(X,Y) = \mathrm{Cov}(Y,X)$
3. $\mathrm{Cov}(aX,bY) = ab \mathrm{Cov}(X,Y)$
4. $\mathrm{Cov}(X_{1}+X_{2},Y) = \mathrm{Cov}(X_{1},Y) + \mathrm{Cov}(X_{2},Y)$
   • 特别地 $\mathrm{Cov}(X+Y,X-Y) = \mathrm{Cov}(X,X) + \mathrm{Cov}(Y,X) - \mathrm{Cov}(X,Y) - \mathrm{Cov}(Y,Y) = DX-DY$
5. $D(X\pm Y) = DX + DY \pm 2\mathrm{Cov}(X,Y)$

例题

1. 设二维随机变量 $(X,Y)$ 的概率分布为 $$\begin{array}{c | ccc} & Y = 0 & Y=1 & Y=2 \\ \hline X = -1 & 0.1 & 0.1 & b\\ X = 1 & a & 0.1 & 0.1 \end{array}$$
   • 若事件 $\{\max\{X,Y\}=2\}$ 和 $\{\min\{X,Y\} = 1\}$ 相互独立，则 $\mathrm{Cov}(X,Y) =$( ) a. -0.6 b. -0.36 c. 0 d. 0.48 $$\begin{array}{ll} \because & \{\max\{X,Y\}=2\} \text{与} \{\min\{X,Y\} = 1\} \text{相互独立} \\ \therefore & \mathbb{P}\{\max\{X,Y\}=2,\min\{X,Y\} = 1\} = \mathbb{P}\{\max\{X,Y\} = 2\}\mathbb{P}\{\min\{X,Y\} = 1\} \\ \therefore & \mathbb{P}\{X=1,Y=2\} = \mathbb{P}\{Y =2\}\mathbb{P}\{X=1, Y \ge 1\} \\ \therefore & 0.1 = (0.1+b)(0.1 + 0.1) \\ \therefore & b = 0.4,a=0.2 \\ \because & \mathrm{Cov}(X,Y) = EXY - EX\cdot EY \\ & = -2\times 0.4 + 2\times 0.1 - (-0.6 + 0.4)(0.2 + 2\times 0.5) \\ & = -0.36 \end{array}$$