Question

Suppose the number of earthquakes occurring in an area approximately follows a Poisson distribution with an average rate of 2 earthquakes every year.

a.) Find the probability that there will be 1 to 3 (inclusive) earthquakes during the next year in this area.

b.) Find the probability that there will be exactly 5 earthquakes during the next 3 year period.

c.) Consider 10 randomly selected years during last century. What is the probability that there will be at least 3 of those years with no earthquake?

Answer 1

Let X denotes the number of earthquakes during the next year in this area.

X ~ Poisson(2)

The probability mass function of X is

a) The probability that there will be 1 to 3 (inclusive) earthquakes during the next year in this area

b) Let Y denotes number of earthquakes during the next 3 year period in this area.

Y ~ Poisson(2*3) or Y ~ Poisson(6)

The probability mass function of Y is

The probability that there will be exactly 5 earthquakes during the next 3 year period

c) The probability that a randomly selected year has no earthquake

Let W denotes the number of years with no earthquake among 10 randomly selected years.

W ~ Binomial(n = 10, p = 0.1353)

The probability mass function of W is

Now,

The probability that there will be at least 3 of those years with no earthquake