a integral of bivariate Gaussian random variables.

I met the following problem when doing estimation and detection homework. The problem asks for a maximum likelihood estimator for (v,$\rho$) of bivariate joint Gaussian, where v is the common variance for Y1 and Y2, rho is correlation coefficient, Y1, Y2 mean zero. Given one pair of observation (y1, y2).

I find that the MLE for $\rho$ is $\hat{\rho}=\frac{2y_1y_2}{y_1^2+y_2^2}$, and $\hat{v}=\frac{y_1^2+y_2^2}{2}$. now I need to show that these estimators are unbiased. The unbiasedness of $\hat{v}$ is easy. Then I need to verify $$E[\hat{\rho}]=\rho$$

I managed to show this by working through the double integral and change to polar coordinate, but the computation is heavy.

Here is my question, is there a easier (neat) solution to show it is unbiased?

This might be helpful. It says that the maximum likelihood estimator is biased and you get an unbiased estimator by replacing $n$ by $n-1$, though. – joriki Apr 25 '12 at 3:01
@joriki: Further, diving by $n$ gives a biased estimator of the covariance when the mean is unknown. When the mean is know (as here) it's unbiased. – leonbloy Apr 25 '12 at 11:20
It's easy to see that $y_1 y_2$ is the MLE-unbiased estimator of $Cov(y_1 y_2)$ and $(y_1^2 +y_2^2)/2$ the MLE-unbiased estimator of $\sigma^2$ and, $\rho = Cov(y_1 y_2)/\sigma^2$. But the ratio of unbiased estimators is not necessarily unbiased. I wonder if there is some elegant reason why that's true in this case. – leonbloy Apr 25 '12 at 17:50