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Chapter 7: Q7.65 (page 357)

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 of5. If next year will be a good year with probability .4or a bad year with probability .6, find the expected value and variance of the number of storms that will occur.

Short Answer

Expert verified

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

Step by step solution

01

Step 1:Concept Introduction

Given Xthe number of storms that will occur next year.

Y=1    next year is a good year2    next year is a bad year

02

Step 2:Explanation

E[X]=E(X∣Y=1)⋅P(Y=1)+E(X∣Y=2)⋅P(Y=2)

=3⋅0.4+5⋅0.6

=1.2+3

=4.2

Var(X)=EX2−(E[X])2

Now,

EX2=EX2∣Y=1⋅P(Y=1)+EX2∣Y=2⋅P(Y=2)

=(3+9)⋅0.4+(5+25)⋅0.6

=12⋅0.4+30⋅0.6

=4.8+18

=22.18

Var(X)=22.8−4.22

=22.8−17.64

=5.16
03

Step 3:Final Answer

TheE[X]=4.2is the expected value andVar(X)=5.16is the variance.

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