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(a) Let N(t) be the number of arrivals in the interval (0, t] for a Poisson process of rate λ. Show that the probability that N(t) is even is [1 + exp(-2λt)]/2. Hint: Look at the power series expansion of exp(-λt) and that of exp(λt), and look at the sum of the two. Compare this with },
Pr{N(t) = n}.
(b) Let -N(t) be the number of even numbered arrivals in (0, t]. Show that -N(t) = N(t)/2 - Iodd(t)/2, where Iodd(t) isa rv that is 1 if N(t) is odd and 0 otherwise.
(c) Use (a) and (b) to find E[-N(t)]. Note that this is m(t) for a renewal process with second-order Erlang inter-renewal intervals.
Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.
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