Let $(X_1,Y_1),\ldots,(X_n,Y_n)$, be a sample from a bi-variate normal distribution with zero means,variances $q_1^2$,$q_2^2$, and correlation $p$. How to Show that
$ r=\frac{\sum_i(X_i \cdot Y_i)}{\sqrt{\sum_i X_i^2 \cdot \sum_i Y_i^2}} $
has Beta distribution with parameters $1/2$ and $(n-1)/2$.
I got this $r$ in solving a testing problem with null hypothesis $p=0$ versus the alternative ($p \not= 0$).