Categories

# Vector of multivariate normal distribution

Let $$\textbf{X} = (X_1, X_2, X_3)^T$$ and $$\textbf{Y} = (Y_1, Y_2, Y_3)^T$$ be independent vectors with multivariate normal distribution, with means $$\mu_X$$ and $$\mu_Y$$ and covariance matrices $$\Sigma_X$$ and $$\Sigma_Y$$ with non-zero determinant. Let $$A_{2 \times 3}$$ and $$B_{3 \times 3}$$ be lineary independent matrices. Find distribution of $$(\textbf{X}^TA^T, \textbf{Y}B^T)^T$$.

This is what I’ve done

Let $$\textbf{Z} = (\textbf{X}^T, \textbf{Y}^T )^T$$. It has multivariate normal distribution with mean $$\mu_{Z}$$ = $$\big(\begin{smallmatrix} \mu_{X}\\ \mu_{Y} \end{smallmatrix}\big)$$ and covariance matrix $$\Sigma_{Z} = \big(\begin{smallmatrix} \Sigma_{X} & 0\\ 0 & \Sigma_Y \end{smallmatrix}\big)$$.

Let C = $$\big(\begin{smallmatrix} A & 0\\ 0 & B \end{smallmatrix}\big)$$.

Then $$(\textbf{X}^TA^T, \textbf{Y}B^T)^T = C\textbf{Z}$$ with multivariate normal distribution $$N(C\mu_{Z}, C\Sigma_{Z}C^T)$$.

Can you tell me if it is good solution or if it isn’t where did I make a mistake?