Question

Suppose that in a certain region of California, earthquakes occur at the average rate of 7 per year.

(a) What is the probability that in exactly three of the next eight years, no earthquakes occur?

(b) What is the **expect**ed number of years to wait until we have a year with exactly 7 earthquakes?

(c) In the next century, how many years would you **expect** to see with more than 10 earthquakes?

Hint: When you see the word "expect" you should expect to use the expected value!

Answer 1

Let Y be a Poisson random variable which denotes the number of earthquakes per year

Mean of Y, = 7

Probability that no earthquake occurs in a random year = P(Y = 0)

= = 0.0009

Now, let X be a binomial random variable which denotes the number of years in the next eight years in which no earthquake occurs

Here, n = 8 and p = 0.0009

(a) The required probability = P(X = 3) =

=

(b) Probability that exactly 7 earthquakes occurs in a random year

= P(Y = 7)

= = 0.149

Thus, expected number of years to wait until we have a year with exactly 7 earthquakes = 1/0.149 = 6.71

(c) Probability that more than 10 earthquakes occur in a random year

= P(Y > 10) = 0.0985

Thus, expected number of years in which there will be more than 10 earthquakes in the next century = P(Y > 10)*100

= 9.85