But you are waiting 2x the average, because the average waiting time should be half of the period.
Consider buses which arrive every ten minutes, exactly. You arrive at the bus stop at a random time. How long should you expect to wait? Not 10 min but 5 min, on average. You are equally likely to arrive at any point during the 10 minutes wait.
Now change the buses to a Poisson process with mean rate 1 bus every 10 minutes. Now you arrive during an interval of average length 20 min, but wait on average 10 min.
