方差

连续

$$Var[X]=E[(X-E[X]{)}^2]=E[X^2]-(E(X){)}^2$$

离散

$Var[X]=E[X^2]-(E[X]{)}^2={\int }_{-\infty }^{\infty }{x}^{2}f(x)dx-{\left({\int }_{-\infty }^{\infty }xf(x)dx\right)}^2$

推导

因为期望运算与实数运算满足结合律，所以有以下推导

$$\begin{array}{rl}& E[(X-E[X]{)}^2]\\ =& E\left[X^2-2E[X]X+(E[X]{)}^2\right]\\ =& E[X^2]-E\left[2E[X]X+(E[X]{)}^2\right]\\ =& E[X^2]-\left[2E[X]E[X]+(E[X]{)}^2\right]\\ =& E[X^2]-(E(X){)}^2\end{array}$$

Tip

方差可看作协方差的随机变量的$X,Y$为同一变量时的特殊情况