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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 -- 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?

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Indeed the term compound is overloaded in statistics with both definitions. I prefer to describe the latter scenario as " a random sum of random variables" rather than a compound distribution and the former as a "continuous mixture" distribution but the term compound is also used and common for both.

The only relationship between the two compound types I am aware of is with a Bernoulli random variable. If the number of summands in the random sum is Bernoulli then the distribution of the sum is $0$ if the Bernoulli was $0$ and $X_1$ if the Bernoulli was $1$. This is equivalently written

$Y\stackrel{d}{=}Z_1X_1\stackrel{d}{=}\sum_{i=1}^{Z_1}X_1.$ Now the distribution of $Y$ after evaluating the sum has a zero-inflated distribution where the zero inflation is $\Pr(Z_1=0)$. This is the same distribution as if you "compounded" because the product representation above can be interpreted as a scale mixture where you mix between a $0$ value and a value given by the scale parameter of the distribution of $X_1$. The marginal distribution here would require compounding. In this case the compounding is not an integral but a finite sum.

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