Let N(t) be a non-homogeneous Poisson process with inten
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Let N(t) be a non-homogeneous Poisson process with intensity ?(t), ? 0; thus the transition probabilities for small h ar
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Let N(t) be a non-homogeneous Poisson process with intensity ?(t), ? 0; thus the transition probabilities for small h are P(N(t + h) ? N(t) = 1) = ?(t)h + o(h), P(N(t + h) = N(t)) = 1 ? ?(t)h + o(h). (a) Write down the forward equations and verify that EsN(t) = exp 0t ?(u) du(s ? 1) . (b) Show that conditional on N(t) = n, the n arrival times are independent having the common density f (u) = ?(u)# 0t ?(u) du, 0 ? u ? t. (c) Now generalize the result of (4.9.10) to the case when recruits arrive at the instants of a non-homogeneous Poisson process. (d) Find the expected population in a Disaster process (4.8.13), when arrivals and disasters appear as non-homogeneous Poisson processes.
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