At a station, buses leave every $t$ seconds carrying all the waiting people. The arrival times of these people is Poisson distributed with mean $\lambda$.
What is the expected waiting time of a passenger?
Attempt
Consider the time interval $I = [t_0, t_0+t]$. The waiting time $W$ of a passenger arriving that $t_1\in I$ is more than $w$ if they arrive in the first $t-w$ seconds of this interval, i.e., $t_1 \in I_w = [t_0, t_0+t-w]$.
Expected number of arrivals in $I$ is $\lambda t$ and that in $I_w$ is $\lambda (t-w)$.
So, a fraction $\frac{t-w}{w}$ of passengers wait for time more than $w$.
Is the last line correct?
PS: This is not the same as the Waiting Time Paradox, where buses arrive at Poisson-distributed times.