Question

Let the number of tropical cyclones per year in the Atlantic Basin (?) have a discrete uniform between 5 to 15; i.e. ? ∈ {5, 6, … , 15} with ?_X(?) = 1 /11. If each cyclone can turn into a major hurricane with probability 0.3; find the expected number of major hurricanes per year (?). Hint: Use the Law of Interated Expectation; ?[?] = ?[?[?|?]].

Answer 1

Let X be the number of tropical cyclones per year. The probability mass function (PMF) of X is

the expectation of X is

The probability that any given tropical cyclone turns into a major hurricane is 0.30

Let Y|X be the number of major hurricanes given X number of tropical cyclones. We can say that Y|X has a Binomial distribution with parameters, number of trials (is the number of tropical cyclones) n=X and success probability (the probability that a tropical cyclone turns into a major hurricane) p=0.30.

Hence we have the following hierarchy.

the expected number of major hurricanes per year (?) is

ans: the expected number of major hurricanes per year (?) is 3