We have seen that if we superposition a fixed number of Poisson process we obtain a new Poisson process. This need however not be true if we superposition a random number of such processes. More precisely, let us superposition N ∈ Fs(p) independent Poisson processes, each with the same intensity λ, where N is independent of the Poisson processes.
(a) Show that the new process is not a Poisson process, e.g., by computing its generating function, or by computing the mean and the variance (which are equal for the Poisson distribution).
(b) Find (nevertheless) the probability that the first occurrence occurs in process number 1.