If x and y are uncorrelated normal variates, x*exp(y) will have a symmetric unimodal distribution with positive excess kurtosis. Has this distribution been named and studied?

  • $\begingroup$ Standard normals $N(0,1)$ or general Normals $N(\mu, \sigma^2)$? And if the latter, do you wish to assume they share the same means and variances or not? $\endgroup$
    – wolfies
    May 6 '14 at 13:34
  • $\begingroup$ I think even the simple case of x = N(0,1) and y = N(0,ysd^2) is interesting. Obviously as ysd approaches zero the product x*exp(y) approaches normality, but has the case of ysd >= 1 been studied? $\endgroup$
    – Fortranner
    May 6 '14 at 13:39

Minxian Yang terms the distribution of x*exp(y) the normal log-normal mixture and studies its properties in the working paper "Normal Log-normal Mixture: Leptokurtosis, Skewness and Applications". He allows for x and y to be correlated.

  • 3
    $\begingroup$ Can you expand on your answer a little. please? Can you say anything about its properties? Could you give a complete reference? Out of curiosity, did you find this after you posted? (It's perfectly valid to post a question you already have an answer to, and to answer your own question, so it doesn't matter either way, but my curiosity is biting me.) $\endgroup$
    – Glen_b
    May 7 '14 at 0:05

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