# Thread: Multivariate distributions that are 'closed' under marginalization?

1. ## Multivariate distributions that are 'closed' under marginalization?

a very, very nice property of the multivariate normal distribution is that any lower-dimensional margins are also multivariate (or univariate, depending on how low you go) normally distributed.

does anyone know any other distributions for which this property is true?

is there some sort of 'general rule' for this? like if a multivariate distribution satisfies such and such properties, then its lower-dimensional marginals are also the same distirbution (but in lower dimensions)?

2. ## Re: Multivariate distributions that are 'closed' under marginalization?

Here's a bivariate Poisson whose marginals are Poisson
http://en.wikipedia.org/wiki/Poisson...n_distribution

From what I remember, there's not just "one" bivariate Poisson, but several people have developed several different bivariate (and trivariate) distributions that have the appropriate properties (I think some of them are listed in the "See Other" of that wikipedia page)

Multivariate probability distributions are a weird thing; I think the Poisson is a good example -- there's not always a clear "multivariate version" of each univariate distribution, but people can develop multivariate distributions that have certain properties that resemble a univariate distribution, but these may not be unique (as in the multivariate Poisson cases)

3. ## Re: Multivariate distributions that are 'closed' under marginalization?

Dason also suggested the multivariate Bernoulli distribution... but after some exhaustive Google searches it seems that the only way to find this out is integrate some dimensions out and hope that whatever results can be expressed as the family of distributions one is interested in.

i was hoping there would be some obscure, hidden theorem out there or some quick check you could do on the PDF... but there aren't none...

NONE!!!

4. ## Re: Multivariate distributions that are 'closed' under marginalization?

well, apparently under a specific partition the multivariate t distribution has also t-distributed marginals

5. ## Re: Multivariate distributions that are 'closed' under marginalization?

Yes, Kotz, that a name I was trying to remember

There's a series of books by Kotz, N. Balakrishnan, and Norman L. Johnson on Univariate and Multivariate continuous and discrete distributions that might be of interest to you; since you're a grad student (if that's up to date) should be no problem getting your hands on them. Lots of info in them you might be interested in, including how to derive obscure distributions

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