Question

The number of winter storms in a good year is a Poisson random variable with mean 3, whereas the number in a bad year is a Poisson random variable with mean 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.

Answer 1

Answer:  Mean = 4.8 and variance = 5.16

Step-by-step explanation:

Since we have given

Let X be the number of storms occur in next year

Y= 1 if the next year is good.

Y=2 if the next year is bad.

Mean for good year = 3

probability for good year = 0.4

Mean for bad year = 5

probability for bad year = 0.6

So, Expected value would be

$E[x]=\sum xp(x)\\\\=3\times 0.4+5\times 0.6\\\\=1.2+3\\=4.2$

Variance of the number of storms that will occur.

$Var[x]=E[x^2]-(E[x])^2$

$E[x^2]=E[x^2|Y=1].P(Y=1)+E[x^2|Y=2].P(Y=2)\\\\=(3+9)\times 0.4+(5+25)\times 0.6\\\\=12\times 0.4+30\times 0.6\\\\=4.8+18\\\\=22.8$

So, Variance would be

$\sigma^2=22.8-(4.2)^2\\\\=5.16$

Hence, Mean = 4.8 and variance = 5.16