Homework Help: Drude Theory of Metals Poisson Distribution Problem

1. Nov 4, 2011

Cider

1. The problem statement, all variables and given/known data

This is the first problem from Ashcroft's Solid-State Physics which I recently picked up due to having far too much free time. The first two parts of the problem relate to the probability that an electron picked at random will have had no collision during the preceding t seconds is $e^{-t/\tau}$ and the following t seconds, as well as that che probability that the time interval between two successive collisions of an electron falls in the range between t and t+dt is $(dt/\tau)e^{-t/\tau}$. I was able to do these, but the following parts I find issue with. The first is to show that the first probability gives an average time back to or up to the next collision is $\tau$, and that the second gives an average time of $\tau$ as well (which I have successfully done).

2. Relevant equations

Listed above.

3. The attempt at a solution
So for the first one, I get the expression $$\langle t \rangle = \int_0^\infty e^{-t/\tau} dt = \tau^2$$
And for the second one, I get the same expression but with the factor of $\tau$ inherent in the equation fixing this square term for it. Obviously throwing this factor into the first equation will fix it, but I'm having trouble motivating any reason for doing so.

The final part asks to explain why the fact that the first part gives an average time between collisions of $2\tau$ does not conflict with the second of $\tau$, and to derive an explicit derivation of the probability distribution of the times between two collisions, but I'd like to focus on the above for the moment.

It seems likely that this is a fairly simple question, so if it's best put in the other board, by all means move it.