5.6.4 Let fX.t/I t  0g and fY.t/I t  0g be independent Poisson processes with respective parameters  and . For a fixed integer

a, let Ta D minft  0IY.t/ D ag be the random time that the Y process first reaches the value

a. Determine PrfX.Ta/ D kg for k D 0;1; : : : :

Hint: First consider  D X.T1/ in the case in which a D 1. Then,  has a geometric distribution. Then, argue that X.Ta/ for general a has the same distribution as the sum of a independent  s and hence has a negative binomial distribution.