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 variance is distributed Log Normally.
Given,
$$X \sim N[\mu_{X},e^{Y}]$$
$$Y \sim N[\mu_{Y},\sigma^2_{Y}]$$
To Determine,
$$f_{X}(x), F_{X}(x), E(X), E(X^{2})$$
Related Question when Mean is Normal
Compound Distribution --- Normal Distribution with 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
Hint:
$X \mid Y \sim \mathcal{N}\left(\mu_{X}, Y\right)$, no?
So $$f_{X}(x) = \int_{-\infty}^{\infty}f_{X\mid Y}(x \mid y)f_{Y}(y)\text{ d}y$$
$F_{X}$ can be easily found from this.
In general, $$\mathbb{E}[g(X)] = \mathbb{E}\left[\mathbb{E}[g(X) \mid Y]\right] = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}g(x)f_{X \mid Y}(x \mid y)f_{Y}(y)\text{ d}x\text{ d}y$$