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I wonder

  1. what families of distributions can satisfy that the sum of their any two random variables still have a distribution in the same family?
  2. what families of distributions can satisfy that the product of their any two random variables still have a distribution in the same family?

My questions arose when I read this reply

  • The sum of normal (Cauchy, Levy) random variables is normal (Cauchy, Levy).

  • The sum of gamma random variables is gamma if the distributions have a common scale parameter.

  • The product of log-normal random variables is log-normal.

I know it is not true that the sum of two normal distributed random variables is still normally distributed. For example, if $X$ is normally distributed, define $Y$ to be $X$ if $|X| > c$ and $Y = −X$ if $|X| < c$, where $c > 0$. Then $X+Y$ is not normally distributed.

I am not sure how to verify if the other claims are right or not.

What can be say about a family of distribution that satisfies the above requirements?

Thanks and regards!

  • 0
    @did: Where is Didier's fun story now?2012-07-02

0 Answers 0