Feynman: 'The “average time until the next collision” is obtained in the usual way:' In The Feynman Lectures on Physics Vol I 43-1 Feynman says 

The “average time until the next collision” is obtained in the usual way: 
  \begin{equation*}
\text{Average time until the next collision} =
\frac{1}{N_0}\int_0^\infty t\,\frac{N(t)\,dt}{\tau}.
\end{equation*}

Since this is "the usual way" I guess I should be ashamed to ask how this equation is established.  Feynman suggest carying out the integration to show that it works, which I did, as shown in the screen scrape below.  But that doesn't really explain the reason for the form Feynman gives.
Is there a more informative way of establishing the formula given by Feynman?

 A: The function $p$, given by
$$p(t) = \frac{1}{\tau}e^{-t/\tau},$$
is the probability density for the time between collisions to be $t$. That is, the probability that the time between collisions is between $t$ and $t+\Delta t$ is given by
$$p(t)\Delta t = \frac{1}{\tau}e^{-t/\tau}\Delta t.$$
Once the probability distribution is established, we compute an average value in the usual way, i.e.
$$\langle t \rangle = \int_0^{\infty}dt~t~p(t)
=\int_0^{\infty}dt~t\frac{1}{\tau}e^{-t/\tau}
=\frac{1}{N_0}\int_0^{\infty}dt~t\frac{N_0}{\tau}e^{-t/\tau}.$$

By "usual way", I'm referencing a standard result from probability theory that the average value of a random variable $x$ with corresponding probability distribution $p(x)$ is given by
$$\langle x \rangle = \int_{-\infty}^{\infty}dx~xp(x).$$
This is likely easier to see in the context of discrete random variables, where the possible values of $x$ are $x_i$, and then the average value of $x$ is the sum over the possible values weighted by their respective probabilities, i.e.
$$\langle x \rangle =\sum_ix_ip(x_i).$$
(The discrete and continous versions are clearly analogous.)
To understand this result, imagine that we performed measurements of $x$ and got the outcome $x_i$ a number $n_i$ of times. Then, the average of this set of measurements is just
$$\langle x \rangle=\frac{\mbox{sum of all measurements}}{\mbox{number of measurements}},$$
which can be written as
$$\langle x \rangle=\frac{1}{N}\sum_ix_in_i,$$
where $N$ is the total number of measurements. Then, we write this as
$$\langle x \rangle=\sum_ix_if_i,$$
where
$$f_i = \frac{n_i}{N},$$
is the frequency of outcome $x_i$.  In the limit as we take many, many measurements, this frequency can be interpreted (under the somewhat problematic frequentist interpretation of probability) as the probability of outcome $x_i$ occurring.
A: Feynman derives an expression for the probability that the molecule survives for a time $t$ without a collision as $P(t)=\exp(-t/τ)$, ie the probability a particle, during a time interval of $t$, not colliding with anything.
At $t$ increases the probability of such an event (no collision) decreases which is what one would expect.  
From the probability function one can find the average time before a collision $\langle t \rangle$.
$$\langle t \rangle =\dfrac{\int ^\infty_0 t \; \exp(t/\tau)\; dt}{\int ^\infty_0  \; \exp(t/\tau)\; dt} = \tau$$
which is the same as that introduced by Feynman except that I have chosen not to add an $N_0$ top and bottom of the equation.
