概率论-基础计算公式

一、均值与方差

$$\begin{array}{ll}D(XY)& =E\{[XY-E(XY){]}^2\}\\ & =E\{X^2{Y}^2-2XYE(XY)+E^2(XY)\}\\ & =E(X^2)E(Y^2)-2E^2(X)E^2(Y)+E^2(X)E^2(Y)\\ & =E(X^2)E(Y^2)-E^2(X)E^2(Y)\end{array}$$

If $E^2(X)=0$ or $E^2(Y)=0$, then we have
$$\begin{array}{ll}D(XY)& =E(X^2)E(Y^2)\\ & =D(X)D(Y)\end{array}$$
where $D(X)=E(X^2)-E^2(X)=E(X^2)$.

---