题目

Problem

The cumulative distribution function of the discrete random variable $W$ , which takes only the values $6$ , $7$ and $8$ , is given by

F(w)=\frac{(w+3)(w-1)}{77} \quad\text{for }w=6,7,8

Find $\operatorname{E}(W)$

(4)

解答

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这是离散型随机变量的 cumulative distribution function。先用 $F(6)$ 、 $F(7)$ 和 $F(8)=1$ 还原每个点的概率，再求期望。

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\begin{align*} F(6)=\frac{(6+3)(6-1)}{77}=\frac{45}{77}. \end{align*}

\begin{align*} P(W=6)=\frac{45}{77}. \end{align*}

Also,

\begin{align*} F(7)=\frac{(7+3)(7-1)}{77}=\frac{60}{77}. \end{align*}

Therefore

\begin{align*} P(W=7)=F(7)-F(6)=\frac{60}{77}-\frac{45}{77}=\frac{15}{77}. \end{align*}

Finally,

\begin{align*} P(W=8)=1-F(7)=1-\frac{60}{77}=\frac{17}{77}. \end{align*}

Hence

\begin{align*} \operatorname{E}(W) =&\,6\cdot\frac{45}{77} +7\cdot\frac{15}{77} +8\cdot\frac{17}{77}\\[3mm] =&\,\frac{511}{77}\\[3mm] =&\,\frac{73}{11}. \end{align*}