Question

Suppose that the hurricane reeord for the last iOyears at a eertain coastal city in Texas is as follows. Year No. of hurricanes 1961 i 1962 O 1963 O 1964 2 1965 i 1966 O 1967 O 1968 2 1969 1 1970 1 The oceurrenee of hurrieanes can be described by a Poisson process. The maximum wind speed of hurricanes usually shows considerable fluctuation. Suppose that those reeorded at this city can be fitted satisfactorily by a lognormal distribution with mean = 100 ft/see and standard deviation = 20 ri/see. (a) Based on the available data, find the probability that there will be at least i hurrieane in this city in the next 2 years. Ans.0.798. (b) If a strueture in this city is designed for a wind speed of 130 ft/sec. what is the probability that the strueture will be damaged (design wind speed exceeded) by the next hurrieane? Ans. 0.08. (c) What is the probability that there will be at most 2 hurricanes in the next 2 years, and that no structure will be damaged during this period? Ans.0.7/8.

Answer 1

Let X be the number of hurricanes in the city in any given year. X has Poisson distribution with parameter . We can estimate the value of using the sample average.

the probability of X(t)=x, the number of hurricanes in a period t is given by

Let Y be the maximum wind speed recorded in any given hurricane.

the mean of Y is ft/sec and the variance is

We need to find the parameters of the log normal distribution

We know that

solving these 2 we get

Now we know that has a normal distribution with mean and standard deviation

a)  The rate of hurricane for a period of 1 year. The rate for a 2 year period would be . Let X(2) indicate the number of hurricanes in a any random t=2 year period.

the probability that there will be at least 1 hurrieane in this city in the next 2 years, is the probability that X(2)>=1

The probability is given as

the probability that there will be at least 1 hurrieane in this city in the next 2 years is 0.7981

b) the structure will be damaged if the wind speed Y is greater than 130ft/sec.

The probability that the structure will be damaged is the probability that Y>130.

c) the events number of hurricanes is independent of the structural damage due to the hurricane. Hence the probability that there will be at most 2 hurricanes in the next 2 years, and that no structure will be damaged during this period =

(the probability that there will be at most 2 hurricanes in the next 2 years) x (the probability that no structure will be damaged during this period )

the probability that there will be at most 2 hurricanes in the next 2 years = the probability that X(2)<=2

the probability that no structure will be damaged during this period is the probabilty that the wind speed Y is less than 130ft/sec

Now, the probability that there will be at most 2 hurricanes in the next 2 years, and that no structure will be damaged during this period is