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(Solved): Arrivals of passengers at a bus stop form a Poisson process x(t) with rate \lambda per unit time. A ...
Arrivals of passengers at a bus stop form a Poisson process x(t) with rate \lambda
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.
(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)).