HW 6.1: A Poisson process puzzler

This problem is worth 0 points.

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This problem was given to me by Markus Meister and Florian Engert.

Recall that the bus stops in Poissonville have the curious feature that the time when the previous bus arrives has no bearing on when the next one arrives. There is, however, an average waiting time, which we will call \(\tau\) minutes.

You get to the bus station in Poissonville and wonder how long you will have to wait for the next bus. You figure that you will have to wait time \(\tau\) minutes on average. You ask the guy who hangs out at the bus stop how long it has been since the last bus arrived. He says that people ask him that all the time and that on average it has been \(\tau\) minutes. So, the average time between arrivals of buses is \(2\tau\) minutes.

Is this right? Why?