# Compound Distribution — Normal Distribution with Normally Distributed Mean

Could someone please point me to a source or suggest ways in which we can obtain the Distribution, Density Functions, Expected Value, etc. of a Normal Distribution whose mean is distributed Normally.

# Given,

$$X \sim N[Y,\sigma^2_{X}]$$

$$Y \sim N[\mu_{Y},\sigma^2_{Y}]$$

# To Determine,

$$f_{X}(x), F_{X}(x), E(X), E(X^{2})$$

Related Question when Variance is Log Normal

Compound Distribution — Normal Distribution with Log Normally Distributed Variance

Wikipedia Link: This is listed as example at this link without proof. https://en.wikipedia.org/wiki/Compound_probability_distribution#Examples

Related General Question

Starting with the above 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 some source where certain combinations and results therein were given with detailed steps and complete proofs: https://math.stackexchange.com/questions/1614212/compound-distributions-basic-techniques-and-key-general-results-from-first-p

• See comment there. – Did Jan 22 '16 at 7:11