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Could someone please point me to a source with notation, terminology, key results and basic techniques to approach compound distributions?

Definition

Compound probability distribution is the probability distribution that results from assuming that a random variable is distributed according to some parametrized distribution, with the parameters of that distribution being assumed to be themselves random variables: https://en.wikipedia.org/wiki/Compound_probability_distribution

Starting with the below special case, it quickly becomes apparent there are many combinations possible. Hence was wondering if there were general techniques to derive the density, distribution function, expected value, higher moments, conditional expectations etc. of compound distributions and any source where the more common combinations and key results therein are given with detailed steps and complete proofs from first principles.

Special Case Related Question

Compound Distribution --- Uniform Distribution with Normally Distributed Parameters

Please note, this is a cross posting on mathematics and statistics forum.

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