Question

It is proposed to model the onset of hurricanes anywhere in the Gulf of Mexico as a Poisson process. The rate of occurrence, however, depends strongly on the month of the year. Specifically, we assume that hurricanes occur in only three months of the year: August, September, and October. During these months, the mean hurricane occurrence rate per month is as follows: Month Mean occurrence rate August 1.0 event per month September 2.0 events per month October 1.0 event per month These occurrence rates are assumed to apply for any given year. Events in different months are assumed to be mutually independent.

(a) If one hurricane occurs in a single year, find the probability that it occurs in October (5 points).

(b) Find the mean hurricane occurrence rate per year. Hint: Because events in different months are independent, the number of events in a year is equal to the number of events in August plus the number of events in September plus the number of events in October (6 points).

(c) Find the probability that at least two hurricanes occur in a single year in the Gulf of Mexico. Hint: Use the mean occurrence rate calculated in part (b) and the assumption that it’s a Poisson process (6 points).

(d) Let X be the random variable that represents the number of hurricanes to occur over the next 50 years. Find the mean and variance of X (6 points).

(e) Assume that any hurricane will strike land with probability p = 0.05, independent of whether any other hurricane also strikes land. Find the probability that at least 2 hurricanes will strike land in the next 50 years. Hint: What is the mean rate of hurricanes striking land in 1 year? In 50 years? (6 points)

(f) Random variable X defined in part (d) can be written as X = X1 + X2 + . . . + X50, where X1, X2, . . . , X50 represent the number of hurricanes occurring in years 1, 2, . . . , 50. If we assume the latter are independent and identically distributed random variables then their sum X will tend to be Gaussian due to the Central Limit Theorem, with its mean and variance equal to those computed in part (d). Using this approximation, find the probability that at least 180 hurricanes occur over the next 50 years (6 points).

Answer 1

a)
X = number of occurence in Month August = pois(1)
Y= number of occurence in Month September = pois(2)
Z = number of occurence in october = pois(1)

X+ Y+Z = pois(4)

P(Z = 1 | X +Y+Z = 1)
= P(Z = 1 ,X = 0 , Y = 0) / P(X+Y+Z = 1)
= e^(-1) * e^(-2) * 1 e^(-1) /(4 e^(-4))
= 0.25

b)
E(X +Y+Z) = 4

c)
P(X+ Y+ Z >= 2) = 1 - P(X +Y+Z =0) - P(X +Y+Z =1)
= 1- e^(-4)(1+4)
= 0.9084

d)
X = S1 + S2+ .. S50
where S = pois(4)
E(X) = 50 E(S) = 50*4 = 200

Var(X) = 50 Var(S) = 50 * 4= 200

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