Given two uncorrelated Gaussian random variables,
X ~ $\mathcal{N}(0,1)$
Y ~ $N(0,1)$
Find $f_{y|x}(y|x)$, $f_{x,y}(x,y)$
If I can find either the conditional probability or the joint probability, I can solve for the other using the relation: $f_{y|x}(y|x) = \frac{f_{x,y}(x,y)} {f_{x}(x)}$.
I know if X and Y are independent, then their jonit pdf will be: $f_{x,y}(x,y) = f_{x}(x)f_{y}(y)$. However, I do not know if X and Y are independent, only that they are uncorrelated. If X and Y are $\mathbf{jointly}$ normal, then they are independent (since they are uncorrelated), but this leads me back to being able to solve for their joint pdf, which I am not sure how to obtain.
Could anyone point me to the right direction of where to start? I see a few previous questions regarding this topic, but it seems they started with the assumption that X and Y were independent, which I don't believe I can make in this case.