Linear combination of two dependent multivariate normal random variables

Suppose we have two vectors of random variables, both are normal, i.e., $X \sim N(\mu_X, \Sigma_X)$ and $Y \sim N(\mu_Y, \Sigma_Y)$. We are interested in the distribution of their linear combination $Z = A X + B Y + C$, where $A$ and $B$ are matrices, $C$ is a vector. If $X$ and $Y$ are independent, $Z \sim N(A \mu_X + B \mu_Y + C, A \Sigma_X A^T + B \Sigma_Y B^T)$. The question is in the dependent case, assuming that we know the correlation of any pair $(X_i, Y_i)$. Thank you.

Best wishes,
Ivan

Answer

In that case, you have to write (with hopefully clear notations)
$$
\left(\begin{matrix}X\\Y \end{matrix}\right) \sim \mathcal{N}\left[ \left(\begin{matrix}\mu_X\\\mu_Y\end{matrix}\right), \Sigma_{X,Y} \right]
$$
(edited: assuming joint normality of $(X,Y)$)
Then
$$
AX+BY=\left(\begin{matrix}A& B \end{matrix}\right)
\left(\begin{matrix}X\\Y \end{matrix}\right)
$$
and
$$
AX+BY+C \sim \mathcal{N}\left[
\left(\begin{matrix}A& B \end{matrix}\right)
\left(\begin{matrix}\mu_X\\\mu_Y\end{matrix}\right) + C,
\left(\begin{matrix}A & B \end{matrix}\right)\Sigma_{X,Y} \left(\begin{matrix}A^T \\ B^T \end{matrix}\right)\right]
$$
i.e.
$$
AX+BY+C \sim \mathcal{N}\left[A\mu_X + B\mu_Y +C,
A\Sigma_{XX}A^T+B\Sigma_{XY}^TA^T+A\Sigma_{XY}B^T+B\Sigma_{YY}B^T
\right]
$$

Attribution
Source : Link , Question Author : Ivan , Answer Author : Xi’an

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