5. (20 pts) We consider a stochastic process N(t) that counts the number of particles arriving at a Geiger counter. Suppose A is an exponential random variable, namely fA(A) = ne-a* for A > 0. Conditional on A = A, N(t) is a Poisson process with rate A. (a) (3 pts) Find P(N(t) = n|A = A) for some t > 0. (b) (6 pts) Using the conditional probability you found in (a), find the marginal distribution of N(t), P(N(t) = n). Hint: Jo e Mix" de = Fail (c) (6 pts) Let 71 denote the time of the arrival of the first particle. Find Fr (t1 ), the cdf of TI. (d) (5 pts) Is N(t) an independent increment process? Prove your result