I want to obtain distribution of mid-range, $(x_{(1)} + x_{(n)})/2$, of an uniform(a, b) random variable. One can use the following transformation.
$M = \frac{X_{(1)} + X_{(n)}}{2}$ and $W = \frac{X_{(n)} - X_{(1)}}{2}$
clearly
$X_{(1)} = M-W$, $X_{(n)} = M + W$ and the jacobian is 2.
The joint distribution of $m, w$ is
$f(m, w) = 2^{n-1}n(n-1)w^{n-2}/(b-a)^n$.
Now i want to integrating with respect to $w$ to find distribution of mid-range. But i cannot determine the limit of $w$. How should determine the limit of that?