协方差

Covariance) $C o v (X, Y$

反映随机变量之间依赖关系的一个数字特征

协方差定义为：｛(一个随机变量减去其期望) 乘以 (另一个随机变量减去其期望)｝的期望，实际上就是随机变量的函数的期望

\begin{aligned} C o v (X, Y) & = E {[X - E (X)] [Y - E (Y)]} \\ = E (X Y) - E (X) E (Y) \end{aligned}

\begin{aligned} C o v (X, Y) & = E [X Y + E (X) E (Y) - X E (Y) - Y E (X)] \\ = E (X Y) + E (X) E (Y) - E (Y) E (X) - E (X) E (Y) \\ = E (X Y) - E (X) E (Y) \end{aligned}

特别的，当 $X, Y$ 独立时， $E (X Y) = E (X) E (Y)$ ，所以协方差 $C o v (X, Y) = 0$

协方差的数值大小受变量单位和量级的影响，因此它不是一个标准化的度量，受 $X, Y$ 本身度量单位的影响, 引出相关系数 $ρ_{X Y}$

\begin{aligned} ρ_{X Y} & = \frac{C o v (X, Y)}{\sqrt{D (X) D (Y)}} \\ C o v (X, Y) & = ρ_{X Y} \sqrt{D (X) D (Y)} \end{aligned}

\begin{aligned} C o v (X, X) = D (X) \\ C o v (X, Y) = C o v (Y, X) \\ C o v (a X, b Y) = a b C o v (X, Y) \\ C o v (C, X) = 0 C 为 常 数 \\ C o v (X_{1} + X_{2}, Y) = C o v (X_{1}, Y) + C o v (X_{2}, Y) \end{aligned}

协方差与方差的关系：

D (X + Y) = D (X) + D (Y) + 2 C o v (X, Y)

当 $X, Y$ 独立时， $D (X + Y) = D (X) + D (Y)$

\begin{aligned} C o v (X_{1} + X_{2}, Y_{1} + Y_{2}) & = C o v (X_{1}, Y_{1} + Y_{2}) + C o v (X_{2}, Y_{1} + Y_{2}) \\ = C o v (X_{1}, Y_{1}) + C o v (X_{1}, Y_{2}) + C o v (X_{2}, Y_{1}) + C o v (X_{2}, Y_{2}) \end{aligned}