# What is the covariance of a combination of multivariate normal distributions?

("Combination" might not be the right word.)

Say we have $$n$$ random variables $$X =[X_1, X_2, ..., X_n]^T$$. We also know that all the random variables have a normal distribution, i.e. $$X_i \sim \mathcal{N}(\mu_i, \Sigma_i)$$ with $$\mu_i$$ mean and $$\Sigma_i$$ covariance. From my lecture notes I have gathered that the expectation of this combination is simply $$E[X] = [E[X_1], E[X_2], ...,E[X_n]]^T = [\mu_1, \mu_2, ..., \mu_n]^T$$, but now I have three questions:

1. Is the combination of multivariate normal distributions itself a normal distribution?
2. Can we find the covariance of this normal distribution?
3. How can we find it?
• You have not specified how the various $X_i$ depend on each other, so we do not have enough information to answer any of the three questions. – angryavian Feb 14 at 16:58
• Hm... I don't know if they are conditionally independent actually, and I'm not sure I can make that assumption. Thank you angryavian. – Mossmyr Feb 15 at 4:14