1.9. Arrivals of passengers at a bus stop form a Poisson process X(t)

with rate A = 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

(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)].

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={ for 0 t 1, elsewhere.