Question

1) Assume that for a recent 41-year period there were 5469 earthquakes which were considered as “strong” earthquakes. Using a Poisson distribution, find the probability that in a given year, there are exactly 150 earthquakes that are considered “strong”.

2) Assume that the mean number of aircraft accidents in the United States is 8.5 per month and that a Poisson distribution applies. Find P(5), the probability of having 5 accidents in a month. Is it unlikely to have a month with 5 accidents?

Answer 1

Solution

Let X = Number of earthquakes in a year that are considered “strong”. Then, we are given X ~ Poisson (λ), where λ = average number of earthquakes in a year = 5469/41 = 133.40 ……………………………………………..….. (A)

Let Y = Number of aircraft accidents in the United States per month. Then, we are given X ~ Poisson (λ), where λ = average number of aircraft accidents in the United States per month = 8.5 …………………………………………….. (B)

Back-up Theory

If a random variable X ~ Poisson(λ), i.e., X has Poisson Distribution with mean λ then

probability mass function (pmf) of X is given by P(X = x) = e ^{– λ}.λ^x/(x!) ……..........................……..(1)

where x = 0, 1, 2, ……. , ∞

Values of p(x) for various values of λ and x can be obtained by using

Excel Function, POISSON(x,Mean,Cumulative) ………….............................................................(1a)

Now to work out the solution,

Q1

Probability that in a given year, there are exactly 150 earthquakes that are considered “strong”.

= P(X = 150)

= 0.0121 [vide (1a) and (A)] ANSWER 1

Q2 Part (a)

Probability of having 5 accidents in a month

= P(X = 5)

= 0.0752 [vide (1a) and (B)] ANSWER 2

Since 0.0752 is not considered small, it is only likely to have a month with 5 accidents ANSWER 3

DONE