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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

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  • $\begingroup$ See comment there. $\endgroup$
    – Did
    Jan 22, 2016 at 7:11

1 Answer 1

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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, $$\mathbb{E}[g(X)] = \mathbb{E}\left[\mathbb{E}[g(X) \mid Y]\right] = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}g(x)f_{X \mid Y}(x \mid y)f_{Y}(y)\text{ d}x\text{ d}y$$

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    $\begingroup$ Just how does one propose to deal with the fact that when $Y \le 0$, $X \mid Y$ is not well-defined? $\endgroup$
    – heropup
    Jan 21, 2016 at 12:50
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    $\begingroup$ @heropup You know, that's an interesting question. I'm not sure. And now that I think about it, I wonder if the OP's question is even valid. I'm looking back at some conjugate prior notes, and I remember now that the additional distribution assumption is done on the mean, rather than the variance. $\endgroup$ Jan 21, 2016 at 12:53
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    $\begingroup$ @Clarinetist Thanks for the pointer. Perhaps we can set the variance to be log normal. $\endgroup$
    – texmex
    Jan 21, 2016 at 12:59
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    $\begingroup$ @user249613 No matter what you end up doing, you're likely going to have a very disgusting expression that you'll likely have to numerically integrate. This isn't going to be pretty. $\endgroup$ Jan 21, 2016 at 13:00
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    $\begingroup$ @user249613 Google normal-normal conjugate prior. $\endgroup$ Jan 21, 2016 at 13:02

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