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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 ($p \not= 0$).

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But clearly, as any correlation coefficient, $-1<r<1$ with probability one, and $\mathbb{P}(r<0) > 0$, while for beta random variable $V$, $\mathbb{P}(V<0) = 0$. Are you sure $r$ is beta distributed ? –  Sasha Mar 30 '12 at 21:32

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