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

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

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