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 (No):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 (No)zo and find P(A). (b) Let 7, be the time of the first earthquake. Find P(7, > 1/2|N1 = 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