Assume that volcanic eruptions in the Auckland volcanic field follow a Poisson process with rate λ =

1 1000 eruptions year−1

.

Eruptions can happen anywhere in the volcanic field. The last eruption, about 600 years ago, created a new island called Rangitoto.

a. Let Y be the time from Rangitoto to the next eruption.

What is the distribution of Y?

b. Let X be the time from now to the next eruption. What is the distribution of X?

c. Find the probability the next eruption will be at least 1500 years from now, in three different ways:

(i) using the PDF of X;

(ii) using the CDF of X;

(iii) using the formula for the probability function of the Poisson distribution.

Verify that you get the same answer from all three methods.

d. Sketch the PDF of X, and mark on your sketch the probability you calculated in part (c).

e. In Chapter 38 we said the probability we would experience an eruption in the next 40 years was about 3.9%. Give a calculation to confirm this figure.