Suppose we have two independent standard normal variables $X$ and $Y$ and lets construct $Z$ such as $Z = \rho X + \sqrt{1-\rho^2 Y}$. Therefore, $Z$ is dependent on $X$ and $Y$. I want to find the distribution of $Z|X, Y$. However, when I try to generate the Multivariate Normal distribution from these three Gaussians, I obtain a singular covariance matrix. I read the Wikipedia article on the Multivariate Normal distribution and it mentions a degenerate case when the covariance matrix is singular, giving the following expression:
$f(\mathbf{x})= \left(\det\nolimits^*(2\pi\boldsymbol\Sigma)\right)^{-\frac{1}{2}}\, e^{ -\frac{1}{2}(\mathbf{x}-\boldsymbol\mu)^{{{\!\mathsf{T}}}} \boldsymbol\Sigma^+(\mathbf{x}-\boldsymbol\mu) }$
where $\boldsymbol\Sigma^+$ is the generalized inverse and $\text{det}*$ is the pseudo-determinant.
So, my question is: is it correct to use this distribution to find $Z|X,Y$? And in this case, can I still say that $Z|X,Y$ has continuous pdf of $\frac{f(x,y,z)}{f(x,y)}$? I lack some knowledge on this topic, so I would also be grateful for some references.