关于概率和期望

若\(x\)是取值\([0,1]\)的随机变量，则有\(\begin{aligned} \int_{0}^{1}Pr[\lambda=x]xdx&=1-\int_{0}^{1}Pr[\lambda\leq x]dx \end{aligned}\)。

容易得出\(\begin{aligned}Pr[\lambda\leq x]=\int_{0}^{x}Pr[\lambda=x]dx\end{aligned}\)

那么由右式可得

\(\begin{aligned}&=1-\int_{0}^{1} \left( \int_{0}^{x}Pr[\lambda=y]dy \right) dx\\&=1-\int_{0}^{1}dy\left(\int_{x}^{1}Pr[\lambda=y]dx\right)\\&=1-\int_{0}^{1}dyPr[\lambda=y](1-y)\\&=1-\int_{0}^{1}Pr[\lambda=y]dy+\int_{0}^{1}Pr[\lambda=y]ydy\\&=\int_{0}^{1}Pr[\lambda=x]xdx\end{aligned}\)

跟换\(\sum\)一样，把两个积分顺序换一下。