1667

已知集合 ${\displaystyle M=\{1,2,3,\cdots ,n\}}$ (${\displaystyle n\in {\mathbb{N}}^{\ast }}$), 从集合 ${\displaystyle M}$ 中随机抽取一个数记为 ${\displaystyle X}$, 再从 ${\displaystyle X,X+1,\cdots ,n}$ 中随机抽取一个数记为 ${\displaystyle Y}$, 则 ${\displaystyle \mathbb{E}[Y]=}$ _____.

根据塔式法则 $$\begin{array}{rl}\mathbb{E}[Y]& =\mathbb{E}[\mathbb{E}[Y|X]]=\mathbb{E}\left[\frac{n+X}{2}\right]\\ & =\frac{n}{2}+\frac{1}{2}\mathbb{E}[X]=\frac{n}{2}+\frac{1}{2}\cdot \frac{n+1}{2}=\frac{3n+1}{4}.\end{array}$$

由题意$$\begin{array}{rl}\mathbb{E}[Y]& =\sum _{k=1}^n\mathbb{P}(X=k)\mathbb{E}[Y|X=k]\\ & =\sum _{k=1}^n\frac{1}{n}\cdot \frac{n+k}{2}=\frac{n}{2}+\frac{1}{2n}\sum _{k=1}^{n}k=\frac{3n+1}{4}.\end{array}$$