Distribution of $X_3$ given that $X_1 + X_2 =1$ for $(X_1,X_2,X_3)$ centered gaussian with given covariance matrix

Let $X=(X_1,X_2,X_3)$ have a multivariate normal distribution with $EX_1 = EX_2 = EX_3 = 0$ and covariance matrix:

$\left( \begin{array}{ccc} 2 & -1 & 1 \\ -1 & 5 & 0 \\ 1 & 0 & 3 \end{array} \right)$

Determine the distribution of $X_3$ given that $X_1 + X_2 =1$

I have a hard time finding examples like this one so I don't know how to deal with them. Some basic explanation would be appreciate if possible. :)

• A standard approach is to compute the covariance matrix of $(X_3,Y)$ with $Y=X_1+X_2$, then to deduce that $X_3=aY+bZ$ for some suitable $(a,b)$, with $Z$ standard normal independent of $Y$. Can you do that? – Did Mar 5 '16 at 10:51
• Not sure if I can. But I'll try: $(X_3,Y)$ = $(1,0,3 : 1, 4,1)$, $Cov(X_3,Y)$ = $\frac{(1-0)*(1-0)+(0-0)*(4-0)+(3-0)(1-0)}{2}$ = 2 – K B Mar 5 '16 at 16:16
• Not sure how to deduce that $X_3 = aY+bZ$ though. – K B Mar 5 '16 at 16:25