
Correlation coefficient
Suppose i have 50 pairs that are independent X,Y. The observations are paired. I have the following estimator for the correlation coefficient.
$\displaystyle r = \frac{\sum ^n _{i=1}(X_i  \bar{X})(Y_i  \bar{Y})}{\sqrt{\sum ^n _{i=1}(X_i  \bar{X})^2} \sqrt{\sum ^n _{i=1}(Y_i  \bar{Y})^2}}.$
How can I estimate the distribution of this estimator without knowing anything about the bivariate distriubution?

$\displaystyle r\sqrt{\frac{n2}{1r^2}}$ follows a $\displaystyle t$ distribution with $\displaystyle n2$ df.

Thanks for answer. How can I show that? Is it possible to derive it?