例 4.1.1 (二项分布的数学期望)

设 $X\sim B\left(n,p\right)$, 试求 $E\left[X\right]$.

解

$$\begin{array}{rl}E[X]& =\sum _{k=0}^{n}k\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}\\ & =\sum _{k=1}^{n}k\frac{n!}{k!(n-k)!}{p}^k(1-p{)}^{n-k}\\ & =np\sum _{k=1}^n\frac{(n-1)!}{(k-1)!(n-1-(k-1))!}{p}^{k-1}(1-p{)}^{n-1-(k-1)}\\ & =np\sum _{l=0}^{n-1}\left(\genfrac{}{}{0ex}{}{n-1}{l}\right)p^l(1-p{)}^{n-1-l}\\ & =np(p+(1-p){)}^{n-1}\\ & =np\text{.}\end{array}$$

这说明 $n$ 重独立伯努利试验中成功的平均次数为 $np$.