Poisson Statistics in Solid State Physics

[blue2004STi]

Homework Statement

In the Drude model the probability of an electron having a collision in an infinitesimal time interval dt is given by dt/$$\tau$$.
(a) Show that an electron picked at random at a given moment will have no collisions during the next t seconds with probability e^-t/$$\tau$$.
(b) Show that the 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/$$\tau$$)}
(c) Show as a consequence of a) that at any moment the mean time up to the next collision averaged over alll electrons is $$\tau$$.
(d) Show that as a consequence of b) that the mean time between successive collisions is $$\tau$$.

Homework Equations

Probability of a collision per unit time = t/$$\tau$$
Poisson Distribution of Random Variables, Poisson(k,$$\lambda$$)= ($$\lambda$$^ke^-dt/$$\tau$$)/k!

The Attempt at a Solution

So I proved part (a) by using the Poisson Distribution of RV's. Part (b) I tried to do the same thing as part (a), but for the time interval I used (t+dt)-t which gave me a lambda of dt/$$\tau$$. Then I used k = 1 and went from there and it worked until the exponent where I got e^-dt/$$\tau$$ rather than e^-t/$$\tau$$. Part (c) and (d) are where I get lost and have no clue of what to do.

 

[blue2004STi]

After discussing this with my professor he said that to solve this problem you don't have to use the Poisson distribution, that being said he also told me that "if you ask the distribution the right questions" that it can be solved using the distribution. Not sure what this means in terms of how to use it... I don't understand how to approach the problem other than using the distribution and for part b) apparently I was wrong originally, but I'm not exactly sure what to "ask" the distribution.

If C is the time between collisions, I know that I want to know P(t < C < t + dt)...now this is where I get stuck, I'm not entirely sure where to go from here...Thoughts?

Thanks,

Matt