If we have an arrival of say passengers to a bus stop where at time $t$ the bus arrives empty and leaves immediately. Where the arrival of passengers arrive with rate $\lambda$ per hour.
$P(S_n \le t) = P(N(t) \ge n) $ where $S_n$ is the sum of arrivals.
We get that $E[N(t)] = t\lambda$.
Inter-arrival times are Exponential with mean $\frac{1}{\lambda}$.
So we have $E[S_n] = \frac{n}{\lambda}$.
If $T$ is the total waiting time we get $E[T] = \frac{\lambda t^2}{2}$.
Now if we have the case there $N(t) = 3$ and we want to know the expected waiting time of of the 3rd/last passenger: this wil be $E[T_3] = \frac{3t}{4}$ as follows from Expected time of third passenger
Now, $E[S_3] = \frac{3}{\lambda}$ so the expected arrival time is the above. intuitively it seems the expected waiting time would then be $E[t-S_3] = t - \frac{3}{\lambda}$ however this is incorrect as the previous result above is the correct solution.
Herein lies my question, why is expected arrival time dependent upon $\lambda$ yet expected waiting time is not? Since total time - arrival time = waiting time.