In a course on Probability Theory, I encountered the following problem on moment generating functions for multivariate random variables:
If $(X,Y,Z)$ is a multivariate normal random variable, with moment generating function:
$$ M = \exp(_1 −2_3 +_1^2 +_1_2 +3_2^2/2 −t_1t_3 +5t_3^2/2) $$ We want to find $a$ such that $X$ and $Y+aX$ are independent.
I tried replacing $t_2$ with $(t_2+at_1)$ and letting $t_3=0$ but I’m not sure if that is correct or where to go from here?
In doing the above, this leads to another form of the MGF, however, I am unsure of how this helps us in any way to find the value of $a$ that guarantees independence. Of course, this substitution specifies another example of the multivariate normal distribution, but I'm unsure if this is even relevant to the question of independence.
I would be grateful for some clarity here.