Consider a counting process in which the rate is a rv η with probability density fη(λ) = αe-αλ for λ > 0. Conditional on a given sample value λ for the rate, the counting process is a Poisson process of rate λ (i.e., nature first chooses a sample value λ and then generates a sample path of a Poisson process of that rate λ).
(a) What is Pr{N(t)=n | η=λ}, where N(t) is the number of arrivals in the interval (0, t] for some given t > 0?
(b) Show that Pr{N(t)=n}, the unconditional PMF for N(t), is given by αtn Pr{N(t)=n} = (tα)n+1 .
(c) Find fη(λ | N(t)=n), the density of λ conditional on N(t)=n.
(d) Find E [η | N(t)=n] and interpret your result for very small t with n = 0 and for very large t with n large.
(e) Find E [η | N(t)=n, S1, S2, ... , Sn]. Hint: Consider the distribution of S1, ... , Sn conditional on N(t) and η. Find E [η | N(t)=n, N(τ )=m] for some τ t.
Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.