My question is related to the correlation between random variables X and Y, where $(X,Y)$ is bivariate normal. My understanding is as follows.

The correlation coefficient is $\rho=\dfrac{\operatorname{Cov}(X,Y)}{\sqrt{\operatorname{Var}(X)\operatorname{Var}(Y)}}$.

Taking samples from $(X,Y)$, the sample correlation coefficient is $R=\dfrac{\sum(X_i-\overline{X})(Y_i-\overline{Y})}{\sqrt{\sum(X_i-\overline{X})^2\sum(Y_i-\overline{Y})^2}}$.

My question is:

Is $R$ an unbiased estimator for $\rho$? If yes, can you guide me to a proof?

Remark: Following the advice in the comment, I split the question into two.

  • $\begingroup$ This is a set of related questions that should br separate. Please pick one to keep, and ask the other separately. Keeping the background similar and linking to the other is a good idea though. $\endgroup$ – Nij Jul 16 '16 at 22:47
  • $\begingroup$ please merge your accounts $\endgroup$ – Glen_b Jul 17 '16 at 6:07

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