Bivariate Normal Transformation

Question

I have $X, Y$ governed by a Bivariate Normal($\mu_{X} = 0, \mu_{y} = 0, \sigma^2_{x} = 1, \sigma^2_{y} = 1, \rho=\rho$). $H, J$ are from the same joint distribution as $X,Y$ and are independent. If $G = cX + (H - aJ)$, what is the joint distribution $f_{X,G}(x,g)$? I am experiencing troubles with the transformation aspect of this.

Answer 1

The random vector $U:=(X,G)^T$ is a linear transformation of the random vector $V:=(X,Y,H,J)^T$, specifically $$ U:=\begin{pmatrix}X\\G\end{pmatrix}= \begin{pmatrix}1&0&0&0\\c&0&1&-a\end{pmatrix} \begin{pmatrix}X\\Y\\H\\J\end{pmatrix}=BV, $$ where we write $B$ for the $2\times 4$ matrix of constants. Since $V$ has multivariate Gaussian distribution, it follows that $U$ also has Gaussian distribution. It remains to determine the mean vector and covariance matrix for $U$. Clearly $U$ has mean vector zero, so the covariance matrix for $U$ has the simple form $$ \operatorname{Var}(U)=E(UU^T)=E[(BV)(BV)^T]=E(BVV^TB^T)=BE(VV^T) B^T=B\Sigma B^T $$ where $\Sigma$ is the covariance matrix for $V$. Now plug in.