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It is well-known that a sum of normal r.v.'s is another normal r.v., and a sum of log-normal r.v.'s can be accurately approximated with a log-normal r.v. But what can we say if we have a mixture of both types is the sum? Is there any other approximation?

The second question is regarding the product. Again, it is not a secret that a product of log-normal r.v.'s is another log-normal r.v., and a product of normal r.v.'s is already not that straight-forward. But can one draw any conclusion regardig the product of normal and log-normal r.v.'s?

Thanks, Ivan

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A sum of log-normal random variables can be accurately approximated with a log-normal random variables, but only at its right-tail. This shows the meaning of approximate should be made much more specific here. – Did Jun 18 '12 at 9:18
@Ivan, was my answer below of any help? – Justin Jun 19 '12 at 16:35
@justin, yes, thanks! – Ivan Jun 19 '12 at 19:39
glad I could help...I've got lots of helpful answers from others so it's nice when I can actually give back :) – Justin Jun 19 '12 at 19:40
up vote 1 down vote accepted

For the second question, there is a significant amount of research on what is called the "Normal Log-normal Mixture" distribution. (NLN)

The form of the RV is:

$x=e^{\nu /2}\varphi $

Where $\nu$ and $\varphi$ are random variables satisfying: $\begin{bmatrix} \varphi\\ \nu \end{bmatrix} \sim N\left ( \begin{bmatrix} 0\\0\\ \end{bmatrix},\begin{bmatrix} 1 &\rho \sigma \\ \rho\sigma &\sigma^2 \end{bmatrix} \right )$

There are a number of extensions from here, but this is a general beginning point to products of these two distributions.

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