# In Example 7.20, show that if F is exponential with rate , then Average Number Waiting =

## Question:

In Example 7.20, show that if F is exponential with rate μ, then Average Number Waiting = E[N]

That is, when buses arrive according to a Poisson process, the average number of people waiting at the stop, averaged over all time, is equal to the average number of passengers waiting when a bus arrives. This may seem counterintuitive because the number of people waiting when the bus arrives is at least as large as the number waiting at any time in that cycle.

**(b)** Can you think of an inspection paradox type explanation for how such a result could be possible?

**(c)** Explain how this result follows from the PASTA principle.

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