例 4.2.5 (随机个随机变量的和)

设

• $X_1,X_2,\cdots$ 为独立同分布随机变量序列, 数学期望为 $\mu$,
• $N$ 为非负整数值随机变量,数学期望存在,
  • 且与 $X_1,X_2,\cdots$ 独立,

试求 $E\left[\sum _{k=1}^N{X}_k\right]$.

解

$$\begin{array}{rl}& E\left[\sum _{k=1}^N{X}_k\right]\\ & =E\left(E\left[\sum _{k=1}^N{X}_k\mid N\right]\right)\\ & =\sum _{n=0}^{\infty }E\left[\sum _{k=1}^N{X}_k\mid N=n\right]\cdot P\left(N=n\right)\\ & =\sum _{n=0}^{\infty }E\left[\sum _{k=1}^n{X}_k\right]\cdot P\left(N=n\right)\\ & =\sum _{n=0}^{\infty }n\mu \cdot P\left(N=n\right)\\ & =\mu \cdot E\left[N\right]\text{.}\end{array}$$