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I have 3 questions to ask:

  1. Do 2 independent normal random variables always have a bivariate normal distribution?
  2. If we have a bivariate normal distribution, is it necessary that the marginals will have normal distribution?
  3. Can we make a bivariate joint distribution from any two random variables?
Avishkar683
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  • Does this answer your question? [Is it possible to have a pair of Gaussian random variables for which the joint distribution is not Gaussian?](https://stats.stackexchange.com/questions/30159/is-it-possible-to-have-a-pair-of-gaussian-random-variables-for-which-the-joint-d) – Peter O. Dec 31 '20 at 17:19
  • You can find answers to all three of these questions elsewhere on the site. – whuber Dec 31 '20 at 17:58

1 Answers1

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  1. If the two random variables are independent, than they together form a bivariate normal distribution with zeroes everywhere in the covariance matrix except the diagonal, since they are uncorrelated.
  2. If random variable is bivariate or multivariate normal, than all the marginals are also normal.
  3. As said above, multivariate normal has univariate normal marginals, so they cannot be “any” variables.

Wikipedia has pretty good article on multivariate normal, you can read if for more details. You can also check other questions tagged as . Especially, you can see the Is it possible to have a pair of Gaussian random variables for which the joint distribution is not Gaussian? thread that asks the opposite, but related question.

Tim
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