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\sect{Average distance between two particles in a box}
In a one dimensional box with length ${L}$, two particles have
random positions $x_1$, $x_2$ . The particles do not know about each other.
The probability function for finding a particle in a specific location in
the box is uniform. Let ${r = x_1 - x_2}$ be the relative distance
of the particles. Find ${\langle\hat{r}\rangle}$ and the dispertion ${\sigma_{r}}$
as follows:
\Dn
(1) By using theorems for "summing" the expectation values and variances of independent variables.
(2) By calculating the probability function ${f\left(r\right)dr={P}\left(r<\hat{r}