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2. Let X be a random variable taking var lues in E = {1, 2, 3, ...} with distribution P{ X = 1} = Ti for all i E E. For each i E E, let { Nt(i) : t 2 0} be a Poisson process with rate 1(2). Suppose the processes { Nt (1) : t 2 0}, {Nt(2) : t > 0}, ... are independent of each other and independent of X. Define for each t 2 0 Nt = Nt(X). a. Compute P{Nt = k} for k = 0, 1, 2, .... b. Does { Nt : t > 0} have independent increments? Justify your