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 (actually, the mean of the log of the variable) is distributed Log Normally.


$$X = e^{W} $$

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

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

To Determine,

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

Related Question when Mean is Normal:

Compound Distribution --- Log 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

  • $\begingroup$ You are posting a large number of these parameter mix distribution problems, ... so many that working them all out by hand will keep you busy for a while. I suggest you get a computer algebra system, and once you know how to do one, you can do them all ... assuming closed-form solutions exist, and the software can find it. $\endgroup$ – wolfies Jan 21 '16 at 13:27
  • $\begingroup$ @wolfies … Thanks for your attention to this. Indeed, I started trying to get a generic technique; but could find none and received suggestions to post specific questions. Could you please elaborate on the computer algebra system that can help in this regard? $\endgroup$ – texmex Jan 21 '16 at 13:31
  • $\begingroup$ Let me suggest to stop flooding the site with minor variations of the same Ur-question and to grab a textbook covering the basics of 1. distributions, 2. normal distributions, 3. expectations of random variables. In the present case, what the hypothesis says is that $e^X=Y+\sigma_XZ$ and $e^Y=\mu_Y+\sigma_YT$ where $Z$ and $T$ are standard normal and are probably independent (although this is not mentioned in the question, but should be). Everything will be deduced from this. To reach $E(X)$ for example, one would note ... $\endgroup$ – Did Jan 22 '16 at 7:08
  • $\begingroup$ ... that the identities above imply $X=\log(\sigma_XZ+\log(\mu_Y+\sigma_YT))$ hence $$E(X)=\iint_{\mathbb R^2}\log(\sigma_Xz+\log(\mu_Y+\sigma_Yt))\frac{e^{-(z^2+t^2)/2}}{2\pi}dzdt.$$ ...Except that, if one keeps one's eyes open, one readily sees that the hypothesis that $e^Y$ should be normal is impossible, absurd, monstrous, whatever you like to call it, since nondegenerate normal random variables are never always positive. To sum up, less questions, but with an ounce of thought in them, please. $\endgroup$ – Did Jan 22 '16 at 7:09
  • $\begingroup$ @Did .. Thanks for this pointer. It is a typo, I have fixed it. $Y$ is normal, but the mean of $X$ is log normal or distributed as $e^{Y}$. The wording of the question was correct, I presume, but this notation was not. If any other issues, please let me know. $\endgroup$ – texmex Jan 22 '16 at 10:25

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