Correlation coefficient

**Zogru11** · March 22nd 2011, 12:49 AM

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?

**Sambit** · March 22nd 2011, 04:50 AM

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

**Zogru11** · March 22nd 2011, 04:54 AM

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

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