Shocks occur according to a Poisson process with rate λ, and each shock independently causes a certain system to fail with probability p. Let T denote the time at which the system fails and let N denote the number of shocks that it takes.
(a) Find the conditional distribution of T given that N = n.
(b) Calculate the conditional distribution of N, given that T = t, and notice that it is distributed as 1 plus a Poisson random variable with mean λ(1 - p)t.
(c) Explain how the result in part (b) could have been obtained without any calculations.