Q7.65 - A First Course in Probability

Question

The number of winter storms in a good year is a Poisson random variable with a mean of $3$, whereas the number in a bad year is a Poisson random variable with a mean of$5$. If next year will be a good year with probability .$4$ or a bad year with probability $.6$, find the expected value and variance of the number of storms that will occur.

Step-by-Step Solution

Answer

Expected value, $E\left[X\right]=4.2$ and variance,$Var\left(X\right)=5.16$.

Step 1:Concept Introduction

Given $X$ the number of storms that will occur next year.

$Y=\left\{\begin{array}{l}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ next year is a good year }\\ 2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ next year is a bad year }\end{array}\right.$

Step 2:Explanation

$E\left[X\right]=E(X\mid Y=1)\cdot P(Y=1)+E(X\mid Y=2)\cdot P(Y=2)$

$=3\cdot 0.4+5\cdot 0.6$

$=1.2+3$

$=4.2$

$\mathrm{Var}\left(X\right)=E\left[X^2\right]-\left(E\right[X]{)}^2$

Now,

$E\left[X^2\right]=E\left(X^2\mid Y=1\right)\cdot P(Y=1)+E\left(X^2\mid Y=2\right)\cdot P(Y=2)$

$=(3+9)\cdot 0.4+(5+25)\cdot 0.6$

$=12\cdot 0.4+30\cdot 0.6$

$=4.8+18$

 $=22.18$

$\mathrm{Var}\left(X\right)=22.8-{4.2}^2$

$=22.8-17.64$

$=5.16$

Step 3:Final Answer

The $E\left[X\right]=4.2$ is the expected value and $\mathrm{Var}\left(X\right)=5.16$ is the variance.