I wonder if I have such a condition, like I know the joint distirbution of: (X, aY+b), where a and b are constant. Is it possible for me to calculate the joint distribution of (X,Y) based on (X, aY+b)? If yes, my idea is:
$(X,aY+b) \rightarrow N((\mu1,\mu2), \begin{pmatrix}\sigma_1^2 & \operatorname{cov}_{1,2}\\ \operatorname{cov}_{1,2} & \sigma_2^2\end{pmatrix})$ So the new variance of Y is Y/a^2, I only need to update my covariance matrix here I believe.
Is it correct? Thanks.