Question 1 (12 marks) Let Me be the number of earthquakes observed in New Zealand (NZ) up to time t (time units in this question are always years). We will assume that (N):20 is a Poisson process with rate 5 (per year), and that the magnitude of each earthquake is Exponentially distributed with mean 2, independent of all other earthquakes. (a) Let A be the event that there are no earthquakes in the second half year. Express A as an event in terms of (N)zen and find P(A). (b) Let 7, be the time of the first earthquake. Find P(7, > 1/2|M1 = 2). (c) What is the distribution of the number of earthquakes in NZ of magnitude 6 or more in the first 10 years? (d) Suppose that there are exactly 7 earthquakes in each of the first 3 years. What is the probability that there are also 7 earthquakes in the 4th year? (e) Find the distribution function of the magnitude of the largest earthquake in the first year (if there is no earthquake this is defined to be (). (f) Give a reason why modelling earthquake occurrences as a Poisson process may be unre- alistic