Passengers arrive at a railway station according to a homogeneous Poisson process with rate λ. At the beginning (time 0), there are no passengers at the station. The train departs at time T. Denote by W be the total waiting time of all passengers arrived up to the departure of the train. Compute E(W).

My solution:

Let $S_i$ be the arriving time of passenger $i$.

$E(W) = E(E(W|N(T)) = E(E(T\cdot N(T) - \sum_{i=1}^{N(T)} S_i))$

$= T\cdot \lambda T - E(\sum_{i=1}^{N(T)} ES_i)$

$= \lambda T^2 - E(\sum_{i=1}^{N(T)} i/\lambda)$

$ = \lambda T^2 - \frac{1}{\lambda}E(1+ 2+ ... + N(T))$

$ = \lambda T^2 - \frac{1}{2\lambda}E((1+N(T))N(T))$

$ = \lambda T^2 - \frac{1}{2\lambda}(EN(T) + (EN(T))^2)$

$ = \lambda T^2 - \frac{1}{2\lambda}(\lambda T^2 + \lambda T^2+(\lambda T^2)^2)$

$ = \lambda T^2 - T - \lambda T^2/2$

$ = \lambda T^2/2 - T$.

I know the answer is wrong but cannot find my mistake.

  • $\begingroup$ Do you have the correct answer? $\endgroup$ – gunes Apr 11 at 17:20

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