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.