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Poisson Processes 231 5.1.6 Let {X(t); t 0} be a Poisson process of rate . For s, t > 0, determine the conditional distribution of X(t), given that X(t + s) = n. 5.1.7 Shocks occur to a system according to a Poisson process of rate . Suppose that the system survives each shock with probability , independently of other shocks, so that its probability of surviving k shocks is k . What is the proba- bility that the system is surviving at time t? 5.1.8 Find the probability Pr{X(t) = 1, 3, 5, . . .} that a Poisson process having rate is odd. 5.1.9 Arrivals of passengers at a bus stop form a Poisson process X(t) with rate = 2 per unit time. Assume that a bus departed at time t = 0 leaving no customers behind. Let T denote the arrival time of the next bus. Then, the number of passengers present when it arrives is X(T). Suppose that the bus arrival time T is independent of the Poisson process and that T has the uniform probability density function f T (t) = 1 for 0 t 1, 0 elsewhere. (a) Determine the conditional moments E[X(T)|T = t] and E {X(T)} 2 |T = t . (b) Determine the mean E[X(T)] and variance Var[X(T)]. 5.1.10 Customers arrive at a facility at random according to a Poisson process of rate . There is a waiting time cost of c per customer per unit time. The cus-