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("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?
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  • $\begingroup$ 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. $\endgroup$
    – angryavian
    Feb 14, 2019 at 16:58
  • $\begingroup$ Hm... I don't know if they are conditionally independent actually, and I'm not sure I can make that assumption. Thank you angryavian. $\endgroup$
    – Mossmyr
    Feb 15, 2019 at 4:14

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