Let $Y$ and $X$ be bivariat normal distributed with expectation vector $\mu=(\mu_Y,\mu_X)^T$ and covariance matrix $\Sigma=\begin{pmatrix}\sigma_Y^2 & p_{XY}\\p_{XY} & \sigma_X^2\end{pmatrix}$. Determine the density $f_{X,Y}$.
I know that the density in general form is given by $$ f_{Y,X}(y,x)=\frac{1}{2\pi}\lvert\Sigma\rvert^{-1/2}\exp\left(-\frac{1}{2}\left(\begin{pmatrix}Y \\ X\end{pmatrix}-\mu\right)^T\Sigma^{-1}\left(\begin{pmatrix}Y \\ X\end{pmatrix}-\mu\right)\right). $$
Update
I determined the densitiy as
$$ f_{(Y,X)}(y,x)=\frac{1}{2\pi\sqrt{\sigma_Y^2\sigma_X^2-p_{XY}^2}}\exp\left(-\frac{1}{2(\sigma_Y^2\sigma_X^2-p_{XY}^2)}(\sigma_X^2(y-\mu_Y)^2-2p_{XY}(x-\mu_X)(y-\mu_Y)+\sigma_Y^2(x-\mu_X)^2\right) $$
Can anybody tell me if that is correct?