Consider the $(X_{i1}, X_{i2}), i=1, \ldots,n $, where $X_{i1}, X_{i2}$ follow bivariate normal distribution with correlation $\rho$. Define the sample means as $\bar X_j= \dfrac{1}{n}\sum_{i=1}^n X_{ij}, j=1,2$, and the sample correlation as $\hat \rho = \dfrac{\sum_i (X_{i1}-\bar X_1)(X_{i2}-\bar X_2)}{\sqrt{\sum_i (X_{i1}-\bar X_1)^2\sum_i (X_{i2}-\bar X_2)^2}}$. Is there any result on the joint distribution of $\bar X_1, \bar X_2$ conditioned on the sample correlation coefficient $\hat \rho$?
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$\begingroup$ Correlation is a weird statistic, highly nonlinear in both sample means. I seriously doubt that you can push through any exact conditioning results. $\endgroup$ – StasK Apr 7 '14 at 18:51
