Q. 4.1 - A First Course in Probability

Question

Suppose that the random variable $X$ is equal to the number of hits obtained by a certain baseball player in his next $3$at-bats. If $P\{X=1\}=3,P\{X=2\}=2$and$P\{X=0\}=3P\{X=3\}$, find $E\left[X\right].$

Step-by-Step Solution

Answer

The value of $E\left[X\right]$ is $=1.075.$

Step 1: Given Information

Given in the question that,

$\mathrm{P}(\mathrm{X}=1)=0.3$

$\mathrm{P}(\mathrm{X}=2)=0.2$

$\mathrm{P}(\mathrm{X}=0)=3\mathrm{P}(\mathrm{X}=3)$

Step 2: Solution of the Problem

We know that

$\mathrm{P}(\mathrm{X}=0)+\mathrm{P}(\mathrm{X}=1)+\mathrm{P}(\mathrm{X}=2)+\mathrm{P}(\mathrm{X}=3)=1$

$0.5+4\mathrm{P}(\mathrm{X}=3)=1$

Find the value,

$\mathrm{P}(\mathrm{X}=3)=0.125$

$P(X=0)=3\times 0.125=0.375$

Step 3: Computation of the Value

Find the value

$\begin{array}{ll}{X}_i& {\mathrm{P}}_i\\ 0& 0.375\\ 1& 0.3\\ 2& 0.2\\ 3& 0.125\end{array}$

$\mathrm{E}\left(\mathrm{X}\right)=\sum {\mathrm{x}}_i{p}_i$

$=0\times 0.375+1\times 0.3+2\times 0.2+3\times 0.125$

$=1.075.$

Step 4: Final Answer

The value of $\mathrm{E}\left(\mathrm{X}\right)$ is $1.075 .$$1.075 .$