6.4. Let {X(t); t ? 0} and {Y(t); t ? 0} be independent Poisson processes with respective parameters A and μ. For a fixed integer

a, let T = min { t ? 0; Y(t) =

a) be the random time that the Y process first reaches the value

a. Determine 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 negative binomial distribution.