Joint distribution of two multivariate normal distributions If we define 2 independent variables $Y_1$ and $Y_2$ as follows:
\begin{align} 
Y_1 &= (Y_{11},Y_{12},Y_{13})^T \sim\mathcal N_3(\mu_1,\Sigma_{11}),  \\
Y_2 &= (Y_{21},Y_{22})^T        \sim\mathcal N_2(\mu_2,\Sigma_{22})
\end{align}
where,
\begin{align}
\mu_1 &= (2, 2, 2)^T  &\Sigma_{11} &= 
\left[\begin{array}{ccc}  3 &1 &0 \\ 1 &2 &0 \\ 0 &0 &3  \end{array}\right]  \\
\mu_2 &= (3, 4)^T     &\Sigma_{22} &= 
\left[\begin{array}{cc}  4 &2 \\ 2 &4  \end{array}\right]
\end{align}
Then how can I find the joint distribution of $Y_{11}-Y_{13}+Y_{22}$ and $Y_{21}-Y_{12}$?
I know its a simple question but I could find if it was asked for $Y_1-Y_2$ or something. How am I supposed to solve it when it is like that?
 A: Assuming that the word independent in the opening statement is used
in the way that probabilists use the word and not in the sense of independent
versus dependent variable as is common in regression analysis, the
joint distribution of the five random variables $Y_{11}, Y_{12}, Y_{13}, Y_{21},Y_{22}$ is the product of the joint distributions of
$Y_{11}, Y_{12}, Y_{13}$, and $Y_{21},Y_{22}$, both of which are multivariate
normal. This $5$-variate joint distributions
is also a multivariate normal distribution in which the mean vector is
just the concatenation $(\mu_1, \mu_2)^T$ of the two mean vectors and
the covariance matrix is
$$\Sigma = \left[\begin{matrix}\Sigma_{11} & 0\\0 & \Sigma_{22}\end{matrix}\right].$$
Thus, the joint distribution of $Y_{11}-Y_{13}+Y_{22}$ and $Y_{21}-Y_{12}$
is a bivariate normal distribution which can be found by the standard
methods involving setting up a linear transformation mapping
$(Y_{11}, Y_{12}, Y_{13}, Y_{21},Y_{22})$ to 
$Y_{11}-Y_{13}+Y_{22},Y_{21}-Y_{12})$ and doing matrix calculations. More
simply, the means and variances of $Y_{11}-Y_{13}+Y_{22}$ and $Y_{21}-Y_{12}$
as well as their covariance can be computed more directly and used
in writing down the mean vector and covariance matrix of this bivariate
normal distribution.
