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 Log Normal Distribution whose mean (mean of the log of the variable) is distributed Normally.
Given,
$$ X = e^{W} $$ $$ W \sim N[Y,\sigma^2_{W}]$$
$$Y \sim N[\mu_{Y},\sigma^2_{Y}]$$
To Determine,
$$f_{X}(x), F_{X}(x), E(X), E(X^{2})$$
Related Question when Mean is also Log Normal:
Compound Distribution --- Log Normal Distribution with Log Normally Distributed Mean
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