## Joint distribution of sum of independent normals [closed]

Suppose we have three independent normally distributed random variables $$X_0 \sim \mathcal{N}(\mu_0, \sigma_0^2),$$ $$X_1 \sim \mathcal{N}(\mu_1, \sigma_1^2),$$ $$X_2 \sim \mathcal{N}(\mu_2, \sigma_2^2).$$

Now, define two new random variables $Y_0 = X_0+X_1$ and $Y_1 = X_1+X_2$.

Let $\vec{Y} = [Y_0 \;\;\; Y_1]^T$

What can we say about the distribution of $\vec{Y}$? Obviously, $Y_0$ and $Y_1$ are not independent. If they were, then $\vec{Y}$ would have been a multivariate normal variable. Any ideas?

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If you get no interest here, try stats.stackexchange.com - and if you do, flag this question for moderator attention to be closed, so we don't duplicate questions. – David Roberts Jan 4 2012 at 0:48
Alternatively, you could try math.stackexchange.com - the question looks like it would belong better there. Clearly both $Y_0$ and $Y_1$ are Gaussian so it seems that the vector $(Y_0,Y_1)$ will be Gaussian with some non-trivial covariance matrix. – Yemon Choi Jan 4 2012 at 0:53
The fact that $Y_0$ and $Y_1$ both are Gaussian does not imply that $\vec{Y}$ is Gaussian. See en.wikipedia.org/wiki/… – eakbas Jan 4 2012 at 1:11
@eakbas: ah yes, I was too hasty with what I said. However, isn't the image of a Gaussian vector under a linear transformation also Gaussian? see en.wikipedia.org/wiki/… – Yemon Choi Jan 4 2012 at 2:35
The question has now been posted at stats.stackexchange.com/questions/20565/… where other people seem to agree with me regarding multivariate Gaussian distributions – Yemon Choi Jan 4 2012 at 3:13