# Composition of Multivariate Gaussians

This question is teasing my intuition for a moment :

$X \tilde{} N(0,S_1)$
$Y|X \tilde{} N(X,S_2)$
Does $Y \tilde{} N(0,S_3)$ with some $S_3 = f(S_1,S_2)$ like (for instance) $S_1+S_2$ ?

What I found is that if $Y$ follows a normal distribution, so this distribution is $N(0,S_1+S_2)$, since :

• $E[Y] = E_X[E_Y[Y|X]] = E_X[X] = 0$
• $E[YY^T] = E_X[E_Y[YY^T|X]] = S_1+S_2$ because $S_2 = E_Y[(Y-X)(Y-X)^T|X] = E_Y[YY^T|X] - XX^T$ and $E_X[XX^T]=S_1$.

But my calculations to show that it is a normal distribution seem to lead nowhere, since my drawings are giving to my intuition some reasons to think it is true.

What I have done until now is using the formula : $$p(y)=\int p(y|x)p(x)dx$$ but multivariate gaussians seems to be too complicated to achieve this calculation.

• If this is a problem for a course or from a textbook, etc, please add the [self-study] tag & read its wiki. Then tell us what you understand thus far & where you're stuck. – gung - Reinstate Monica Jan 2 '15 at 16:21
• This question is more about curiosity than anything else. I did post a question about an exercise I found in a book there but this one is just something I am asking myself about. I am not even sure there is an answer. Is this self-study ? Because if it is, an not just curiosity, why most of the questions aren't tagged [self-study] ? But I can add some details about what I found if you want. – Thrastylon Jan 2 '15 at 16:40
• Curiosity as a motivation does not make a question self-study. However, a question that is essentially the same as a question that comes from a book or course can be self-study, even if the question itself didn't. The topic itself is somewhat fuzzy. If this isn't from a book or a course, you don't have to tag it [self-study]. – gung - Reinstate Monica Jan 2 '15 at 16:49
• However, the question isn't quite clear to me. 1st, in the last line, should "Y" be something else (Z, perhaps)? After all, you already know what Y is by stipulation. 2nd the last line doesn't quite form a grammatical question. Are you asking if Y (Z) could be distributed as N(0, S3)? – gung - Reinstate Monica Jan 2 '15 at 16:51
• No, I want to know the marginal distribution of Y given its conditionnal distribution subject to X. Things about sum or product of gaussian distributions are well known, but as said in the title, I am here interested on their composition. – Thrastylon Jan 2 '15 at 17:06