Question

The number of storms in a good year is a Poisson random variable with mean 3, whereas in a bad year the number is Poisson with mean 5. The probability that next year will be a good year has probability 0.6 and being a bad year has probability 0.4. Find the expected value and variance of the number of storms next year.

Answer 1

Let X be the number of winter storms next year and letY 1 if next year is a good year and Y 0 otherwise. Then, E(XY 1)3 and E(XIY 0) E(X) = E[EXIY )]-E(XlY-1)P(Y-1) + E(XI Y = 0)P(Y = 0) = 3 . 0.6 + 5 . 0.4-3.8. Similarly, var(XY 1) 3 and var(XY 0) 5. We compute: var[EXIY )] = 3"2* 0.6 + 5^2* 0.4-15a, E[var(XY)] 3* 0.6+ 5 *0.4 3.8, and so var(X) E[var(X Y )]+ var[E(X Y )] 19.2 5. So,