# Correlation coefficient

• March 22nd 2011, 01:49 AM
Zogru11
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.

$r = \frac{\sum ^n _{i=1}(X_i - \bar{X})(Y_i - \bar{Y})}{\sqrt{\sum ^n _{i=1}(X_i - \bar{X})^2} \sqrt{\sum ^n _{i=1}(Y_i - \bar{Y})^2}}.$

How can I estimate the distribution of this estimator without knowing anything about the bivariate distriubution?
• March 22nd 2011, 05:50 AM
Sambit
$r\sqrt{\frac{n-2}{1-r^2}}$ follows a $t$ distribution with $n-2$ df.
• March 22nd 2011, 05:54 AM
Zogru11
Thanks for answer. How can I show that? Is it possible to derive it?