6.4. Let {X(t); 1 2 0) and {Y(t); t 2 0) be independent Poisson processes with respective parameters A and M. For a fixed integer a, let T. = min{t 2 0; Y(1) = a) be the random time that the Y process first reaches the value a. Determine Pr(X(T.) = k} for k = 0, 1, .... Hint: First consider & = X(T,) in the case in which a = 1. Then & has a geometric distribution. Then argue that X(T,,) for general a has the same distribution as the sum of a independent &'s and hence has a neg- ative binomial distribution