Arrivals of passengers at a bus stop form a Poisson process {N(t),t ≥ 0} with rate $\lambda$ = 2 per unit of time. Assume that a bus departed at time t = 0 leaving no customers behind. Let T be the arrival time of the next bus. Then, the number of passengers present when it arrives is N(T). Suppose that the bus arrival time T is independent of the Poisson process and that T ∼ U$_{[0,1]}$ (Uniform distribution over [0, 1]). By first conditioning on T, determine the mean and the variance of the number of passengers present when the next bus arrives (i.e., E(N(T)) and Var(N(T))).
$\lambda$ = 2
E(N(T) = ??
Var(N(T)) = ??
I am not sure about the formulas for how to solve for these with the Uniform distribution.