In good years, storms occur according to a Poisson process with rate 3 per unit time, while in other years they occur according to a Poisson process with rate 5 per unit time. Suppose next year will be a good year with probability 0.3. Let N(t) denote the number of storms during the first t time units of next year.

(a) Find P{N(t) = n}.

(b) Is {N(t)} a Poisson process?

(c) Does {N(t)} have stationary increments? Why or why not?

(d) Does it have independent increments? Why or why not?

(e) If next year starts off with three storms by time t = 1, what is the conditional probability it is a good year?