Dear mathoverflow community,

I tried to find such a distribution family that "forms a semiring", but I failed.

For example, it is common knowledge that Gaussian distribution family forms a semigroup: let us independently sample 2 variables then their sum also has Gaussian distribution.

So, does such a parametric distribution family exist that is closed with respect to addition and multiplication simultaneously?

Dear mathoverflow community,

I tried to find such a parametric distribution family that "forms a semiring", but I failed.

For example, it is common knowledge that Gaussian distribution family forms a semigroup: let us independently sample 2 variables then their sum also has Gaussian distribution.

So, does such a parametric distribution family exist that is closed with respect to addition and multiplication simultaneously?