# Poisson process, combined waiting time

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.

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