Distribution of sums and differences of n correlated normal random variables

Question

Suppose $x_1\sim\mathcal N(2,0.5),x_2\sim \mathcal N(2,3),$ and $x_3\sim \mathcal N(2.5,7)$ with correlations $\rho_{(1,2)}=0.3,\rho_{(1,3)}=0.1,$and $\rho_{(2,3)}=0.4.$ What is the distribution of

1. $x_1+x_2+x_3?$

2. $x_1+ (3\times x_2)+x_3?$

3. $x_1+x_2+(0.5\times x_3)?$

   Solutions:

4. The distribution of $x_1+x_2+x_3 =\mathcal{N}(6.5,15.275)$.

5. The distribution of $x_1+(3\times x_2)+x_3=\mathcal{N}(10.5,30.077)$

6. The distribution of $x_1+x_2+(0.5\times x_3)=\mathcal{N}(5.25,8.005)$

   But I want to know what is the distribution of $(2\times x_1)-(3\times x_2)-x_3$ and $x_1+x_2-(2\times x_3)?$

   The distribution of $(2\times x_1)-(3\times x_2)-x_3$ is $\mathcal{N}(-4.5,41.8407).$

The distribution of $x_1 +x_2-(2\times x_3) $ is $\mathcal{N}(-1,24.1544)$

Answer 1

If they're assumed to be jointly normal, a situation which normally should be given to you, we already know that any linear combination of jointly normal RVs is a univariate normal. In that case, we only need to figure out the mean and variance. Your answer for (1) is therefore correct. For the other options you'll apply the following formulas:

$$E[aX+bY+cZ]=a\mu_x+b\mu_y+c\mu_z$$ $$\begin{align}\operatorname{var}(aX+bY+cZ) &=a^2\sigma_x^2+b^2\sigma_y^2+c^2\sigma_z^2+2(ab\rho_{xy}\sigma_x\sigma_y+ac\rho_{xz}\sigma_x\sigma_z+bc\rho_{yz}\sigma_y\sigma_z)\end{align}$$ In (1), you implemented and obtained this for $a=b=c=1$.