Finding independent Gaussian variables from moment generating functions

Question

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:

$$ M = \exp(𝑡_1 −2𝑡_3 +𝑡_1^2 +𝑡_1𝑡_2 +3𝑡_2^2/2 −t_1t_3 +5t_3^2/2) $$
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.

Answer 1

As $(X, Y+aX)$ is multivariate normal, $X$ and $Y+aX$ are independent if and only if they are uncorrelated. Therefore, we want to find the value of $a$ such that the following holds:

$$ \text{Cov}(X, Y+aX) = \text{Cov}(X,Y) + \text{Cov}(X,aX) = \text{Cov}(X,Y)+ a\text{Var}(X) = 0 $$

The specified MGF tells us that:

• $\mu _X = 1$, $\mu _Y = 0$, and $\mu _Z = -2$
• $ \text{Var}(X) = 2$, $\text{Var}(Y) = 3$, and $\text{Var}(Z) = 5$
• $\text{Cov}(X,Y) = 1$, $\text{Cov}(X,Z) = -1$, and $\text{Cov}(Y,Z) = 0$

And so by substitution, we have:

$$ \text{Cov}(X,Y)+ a\text{Var}(X) = 1 +2a = 0 $$

Therefore, in order to guarantee that $X$ and $Y+aX$ are uncorrelated we have that $a = - \frac{1}{2}$.

And since the variables are multivariate gaussian, independence and uncorrelatedness are equivalent and so we are done.