Find the marginal PDF I am having a hard time trying to find the marginal of the following joint pdf.
$$f_{U,V}(u,v) =2 \frac{n!}{\left( \frac{n}{2}-1 \right)!\left( \frac{n}{2}-1 \right)!}\left[(u-\theta)(\theta + 1-2v+u)\right]^{\frac{n}{2}-1}$$
where $ u<v<\frac{u+\theta+1}{2}$ and $\theta < u <\theta+1$.
This was my attempt
$\begin{align*}
f_V (v) &= \int_{\theta}^{\theta + 1} \frac{n!}{\left( \frac{n}{2}-1 \right)!\left( \frac{n}{2}-1 \right)!}\left[(u-\theta)(\theta + 1-2v+u)\right]^{\frac{n}{2}-1} \cdot 2 du\\
&=  \frac{n!}{\left( \frac{n}{2}-1 \right)!\left( \frac{n}{2}-1 \right)!}\int_{\theta}^{\theta + 1}\left[(u-\theta)(\theta + 1-2v+u)\right]^{\frac{n}{2}-1} \cdot 2 du\\
\end{align*}$

Your support is: $\theta\lt u< \theta +1$ and $u< v< (u+\theta+1)/2$.
That is equivalently: $\theta\leq v<\theta+1$ and $\max(\theta,2v-\theta-1)\leq u<v$
