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I am trying to solve the following problem from Sethna's book on statistical mechanics (not homework).

On a Highway, the probability of a car passing in some intervaldtis [itex]\frac{dt}{\tau}[/itex]; [itex]\tau=5min[/itex].

what is the probability distribution of time intervals [itex]\Delta[/itex] between two consecutive cars. and what is the mean of this distribution?

My attempt:

In a previous question, I derived the probability distribution for n cars to pass in an interval T

[itex]\rho_{car}(n)=\frac{1}{n!}(\frac{T}{\tau})^ne^{-\frac{T}{\tau}}[/itex]

which I believe is correct, as the hint in the question said that I should get a Poisson distribution.

now if I input n=1 i get:

[itex]\rho_{car}(1)=\frac{T}{5}e^{-\frac{T}{5}}[/itex]

this is the probability distribution for a single car to pass in a time interval T (at some time t<T).

according to my calculations the mean of this distributions is 50 minutes^2 which really doesn't make sense.

How do you solve this problem?

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# Probability distribution of the time interval between two cars

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