4. Let 73 and Q be independent Poisson processes on [0, 00) having rates /\\ and ,u, respectively. Let (M(t))t20 be the counting process of 73' and let (N (t))t20 be the counting process of Q. Assume that A 7 a and /\\ 7E 0, it 7 0. (a) (6 marks) Which of the following are true? (i) MO) is independent of M(1) + N(1). (ii) M(2) MO) is independent of M(1) + N(1). (iii) M (2) + N (3) is Poisson distributed. (b) (5 marks) What is the probability that there are three points in the M -process before the second N -point arrives? (c) (4 marks) Show that (M (15)),520 can be represented as a continuous- time Markov chain. (Why does the Markov property hold? What is the rate matrix Q for the chain?) ((1) (6 marks) For the matrix Q from part (c), What is etQ? What are the forward and backward equations for the chain