If $(X_i,Y_i), i = 1, \dots, n$ are independently bivariate normal distributed, with mean $(\mu_x , \mu_y)$ and variances $(\sigma_x^2, \sigma^2_y)$ and correlation coefficient $\rho = 0$.

Denote $T = \frac{\sum_i (X_i - \mu_x)(Y_i - \mu_y)} {\sqrt{\sum_i (X_i - \mu_x)^2 \sum_j (Y_j - \mu_y)^2}}$ the sample correlation. How can I show $T^2$ is distributed as a Beta distribution?

I know $(X_i - \mu_x)^2$ is a $\chi^2_1$, but I can't figure out what are the relations between distributions and algebra manipulation to show it is beta distributed.

  • $\begingroup$ We know that density of correlation coefficient $R$ when $\rho=0$ is $$f_R(r)\propto (1-r^2)^{(n-4)/2}\mathbf1_{|r|<1}$$ By a change of variables $Y=R^2$, it follows that $Y\sim \text{Beta}(\frac{1}{2},\frac{n}{2}-1)$. $\endgroup$ – StubbornAtom Feb 10 at 20:52

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