Let $X_1,X_2, \ldots , X_n$ be independent and identically distributed Uniform random variables on the interval (0, a) for a > 0, each having a density function $f(x) = \frac{1}{a}$, $0<x<a$. Let $X_{(1)},X_{(2)}, \ldots , X_{(n)}$ denote the order statistics. The range of the data is defined as $R = X_{(n)}−X_{(1)}$ and the midrange is defined as $V = \frac{1}{2}(X_{(1)}+X_{(n)})$ Derive the joint distribution of $(R, V )$ and deduce the marginal distributions of $R$ and $V.$
Attempt:
First I am attempting to generate the joint distribution of $(V,X_{(1)})$ and then integrate out the $X_{(1)}$.
For the joint distribution of $(V,X_{(1)})$ I have:
$$ f_{v,x_{(1)}}(v,x_{(1)})=2n(n-1)\int_{-\infty}^v [F(2v-x_{(1)})-F(x_{(1)})]^{n-2} f(2v-x_{(1)})f(x_{(1)}) \, dx_{(1)}$$
At this point I have no idea how to integrate out the $x_{(1)}$.
Edit: I have located the joint distribution of the range and mid-range:
$$ f_{r,v}(r,v) = n(n-1) \left[F\left(v+\frac{r}{2}\right) - \left(F\left(v-\frac{r}{2}\right)\right)\right]^{n-2} f\left(v-\frac r 2 \right) f(v+\frac{r}{2})$$
For each marginal distribution, I have to integrate out:
$$ f_r(r) = \int_{-\infty}^\infty n(n-1)[F(v+\frac{r}{2})-(F(v-\frac{r}{2})]^{n-2}f(v-\frac{r}{2})f(v+\frac{r}{2}) \, dv$$
$$ f_v(v) = \int_0^{\infty}n(n-1)[F(v+\frac{r}{2})-(F(v-\frac{r}{2})]^{n-2}f(v-\frac{r}{2})f(v+\frac{r}{2}) \, dr$$