I'm trying to sort out two different uses of the term "compound distribution" and figure out the relationship.


The Wikipedia article on [compound distribution](http://en.wikipedia.org/wiki/Compound_distribution) -- which I wrote -- defines a compound distribution as an infinite mixture, i.e. if $p(x|a)$ is a distribution of type F, and $p(a|b)$ is a distribution of type G, then $p(x|b) = \int\_a p(x|a) p(a|b) da$ is a compound distribution that results from compounding F with G. This is the distribution of prior and posterior predictive distributions in Bayesian statistics.


However, the term "compound distribution" has another meaning as a random sum, i.e. a sum of i.i.d. variables where the number of variables is random.


What's the relation between the two? And am I using "compound distribution" correctly for the first definition?



compound distribution in Bayesian sense vs. compound distribution as random sum?
I can't say in general, but in the actuarial literature, the random sum of random variables is called an *aggregate distribution*, as in aggregate insured losses. Your definition of compound distribution is the one used in insurance.


