题目

Problem

The discrete random variable $X$ takes the values $-1$ , 2, 3, 4 and 7 only.

Given that

P(X=x)=\frac{8-x}{k}

for $x=-1,2,3,4$ and 7

find the value of $E(X)$

(5)

解答

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先用总概率为 1 求 $k$ ，再代入期望公式。不要急着算期望，因为每个概率里都含有 $k$ 。

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The probabilities are

\begin{array}{c|ccccc} x&-1&2&3&4&7\\ \hline P(X=x)&\frac9k&\frac6k&\frac5k&\frac4k&\frac1k \end{array}

Since the probabilities sum to 1,

\begin{align*} \frac{9+6+5+4+1}{k}=1. \end{align*}

\begin{align*} \frac{25}{k}=1, \end{align*}

and therefore

\begin{align*} k=25. \end{align*}

Now

\begin{aligned} E(X) =&\,(-1)\frac9{25}+2\frac6{25}+3\frac5{25} +4\frac4{25}+7\frac1{25}\\ =&\,\frac{-9+12+15+16+7}{25}\\ =&\,\frac{41}{25}. \end{aligned}