Under the model $(X_1, Y_1), \ldots, (X_n, Y_n) \stackrel{i.i.d.}{\sim} N(0, I_2)$ (so that $\rho_{X,Y} = 0$), the sample correlation $\hat{\rho}_{X,Y}$ has density \begin{align*} f_{\hat{\rho}}(r) = \frac{\Gamma\!\left(\frac{n-1}{2}\right)}{\Gamma\!\left(\frac{n-2}{2}\right)} (1 - r^2)^{\frac{n-4}{2}}, \quad -1 \leq r \leq 1. \end{align*}