Section05_相关系数

基本概念

定义

$$\begin{split} & \rho_{XY} = \mathrm{Cov}\bigg(\frac{X-EX}{\sqrt{EX}}, \frac{Y-EY}{\sqrt{EY}}\bigg) \\ = & \frac{\mathrm{Cov}(X,Y)}{\sqrt{EX}\cdot \sqrt{EY}} \\ \Rightarrow & \mathrm{Cov}(X,Y) = \rho_{XY}\cdot \sqrt{DX} \sqrt{DY} \end{split}$$

• 反映 $X,Y$ 线性关系的紧密程度

性质

1. $\vert \rho_{XY} \vert\le 1$
2. $\rho_{XY} = 0$ 称 $X$ 与 $Y$ 不相关(无线性关系)
   • $\Leftrightarrow \mathrm{Cov}(X,Y) = 0$
   • $\Leftrightarrow D(X\pm Y) = DX + DY$
   • $\Leftrightarrow EXY = EX \cdot EY$
3. $\vert \rho_{XY} \vert = 1 \Leftrightarrow \mathbb{P}\{Y = aX + b\} = 1, a\ne 0$
   • $a < 0 \Rightarrow \rho_{XY} = -1$
   • $a > 0 \Rightarrow \rho_{XY} = 1$
4. $X,Y$ 独立(无任何关系) $\substack{\rightarrow\\\nleftarrow}$ $X,Y$ 不相关(无线性关系)
5. $(X,Y) \sim \mathcal{N}(\mu_{1},\mu_{2};\sigma^{2}_{1},\sigma^{2}_{2};\rho)$
   • $\rho = 0 \Leftrightarrow X,Y\text{不相关}\Leftrightarrow X,Y \text{独立}$ (仅对二维正态分布成立)

例题

1. 设二维离散型随机变量的概率分布为 $$\begin{array}{c|ccccc} (X,Y) & (0,0) & (1,1) & (0,2) & (2,0) & (2,2) \\ \hline P & \frac{1}{4} & \frac{1}{3} & \frac{1}{4} & \frac{1}{12} & \frac{1}{12} \end{array}$$
   1. 判断 $X$ 和 $Y$ 是否不相关
   2. 判断 $X$ 和 $Y$ 是否独立
   3. 求 $\mathrm{Cov}(X-Y,Y)$ $$\begin{array}{ll} \because & \mathrm{Cov}(X,Y) = EXY - EX\cdot EY\\ \therefore & \mathrm{Cov}(X,Y) = \frac{1}{3}+4\times \frac{1}{12} - (\frac{1}{3} + 2\times \frac{1}{12} + 2\times \frac{1}{12})(\frac{1}{3} + 2\times \frac{1}{4}+ 2\times \frac{1}{12}) \\ & = \frac{2}{3} - \frac{2}{3} = 0 \\ \therefore & X,Y \text{不相关} \\\\ \because & \begin{array}{c |ccc|c} & Y=0 & Y=1 & Y=2 \\ \hline X = 0 & 1/4 & 0 & 1/4 & 1/2 \\ X = 1 & 0 & 1/3 & 0 & 1/3 \\ X = 2 & 1/12 & 0 & 1/12 & 1/6 \\ \hline & 1/3 & 1/3 & 1/3 \end{array} \\ \therefore & p_{ij} \ne p_{i\cdot}p_{\cdot j} \\ \therefore & X,Y \text{不独立} \\ \\ & \mathrm{Cov}(X-Y,Y) = \mathrm{Cov}(X,Y) - \mathrm{Cov}(Y,Y) \\ & = 0 -DY = [EY]^{2} - EY^{2} = 1 - \frac{5}{3} = - \frac{2}{3} \end{array}$$
2. 将一枚硬币独立重复抛 $n$ 次，以 $X$ 和 $Y$ 分别表示正面向上和反面向上的次数，则 $X$ 与 $Y$ 的相关系数是 (A) A. $-1$ B. $0$ C. $\frac{1}{2}$ D. $1$
3. 设 $X\sim U(0,3)$，$Y$ 服从参数为 $2$ 的泊松分布，且 $X$ 与 $Y$ 的协方差为 $-1$，则 $D(2X-Y+1)=$         $$\begin{array}{ll} & D(2X-Y +1) = 4 DX + DY - 2\mathrm{Cov}(2X,Y) \\ & = 4\times \frac{3^{2}}{12} + 2 - 4\times (-1) \\ & = 9 \end{array}$$
4. 设 $X\sim \mathcal{N}(0,4), Y\sim B(3, \frac{1}{3})$，且 $X$ 与 $Y$ 不相关，则 $D(X-3Y+1) =$         $$\begin{array}{ll} \because & X,Y \text{不相关} \\ \therefore & \mathrm{Cov}(X,Y) = 0 \\ & D(X-3Y+1) = DX + 9DY - 2\mathrm{Cov}(X,-3Y) \\ & = DX + 9DY +6\mathrm{Cov}(X,Y) \\ & = 4 + 9\times 3\times \frac{1}{3}\times \frac{2}{3} = 10 \end{array}$$
5. 设 $X\sim \mathcal{N}(0,1)$ 在 $X = x$ 的条件下 $Y\sim \mathcal{N}(x,1)$，则 $X$ 与 $Y$ 的相关系数为         $$\begin{array}{ll} \because & \text{在}X = x \text{条件下} Y \sim \mathcal{N}(x,1) \\ \therefore & \displaystyle f_{Y\vert X}(y\vert x) = \frac{1}{\sqrt{2\pi}}e^{-\frac{(y-x)^{2}}{2}} \\ \therefore & \displaystyle f(x,y) = f_{X}(x)f_{Y\vert X}(y\vert x) = \frac{1}{\sqrt{2\pi}\sqrt{2\pi}} e^{-\frac{2x^{2} - 2xy + y^{2}}{2}} \\ \because & \text{当}(X,Y)\sim \mathcal{N}(\mu_{1},\mu_{2};\sigma^{2}_{1},\sigma^{2}_{2}; \rho) \text{时} \\ & \displaystyle f(x,y) = \frac{1}{\sqrt{2\pi}\sigma_{1}\sqrt{2\pi}\sigma_{2}\sqrt{1-\rho^{2}}} \times \\ & \displaystyle \exp\bigg\{-\frac{1}{1-\rho^{2}}\bigg[\frac{(x-\mu_{1})^{2}}{2\sigma_{1}^{2}} - \rho\frac{(x-\mu_{1})(y-\mu_{2})}{\sigma_{1}\sigma_{2}} + \frac{(y-\mu_{2})^{2}}{2\sigma_{2}^{2}}\bigg]\bigg\} \\ \because & X\sim \mathcal{0,1} \\ \therefore & \mu_{1} = 0,\sigma_{1} = 1 \\ \because & \displaystyle -\frac{(x-\mu_{1})^{2}}{2\sigma_{1}^{2}(1-\rho^{2})} = -x^{2}\\ \therefore & \displaystyle \frac{x^{2}}{2(1-\rho^{2})} = x^{2} \\ & \displaystyle 1-\rho^{2} = \frac{1}{2} \\ \therefore & \rho = \frac{\sqrt{2}}{2} \\ \end{array}$$