Say $Z \sim N(\mu, \sigma^2)$. I am trying to figure out what is the variance of $Y = exp(Z)$.

The first thing that came to mind was approximate $exp(Z)$ with a second order taylor series, then it boils down to finding the variance of a polynomial. But then what is the covariance term of $(Z, Z^2)$?

  • $\begingroup$ What is $Z$? Do you mean $X$? If yes, then use moment generating function of $X$ i.e. $M_X(t)=E(e^{tX})=e^{t\mu+t^2/2\sigma^2}.$ $\endgroup$
    – Stat
    Apr 10, 2014 at 18:00
  • $\begingroup$ More answers can be found by searching our site for lognormal moments. $\endgroup$
    – whuber
    Apr 10, 2014 at 18:25
  • $\begingroup$ Finding the covariance of $(Z,Z^2)$ does not require introducing $Y=\exp(Z),$ which seems superfluous for that purpose. Are you trying to ask a question about lognormal moments (moments of $Y$), about covariances of polynomials in a normal variable (such as $Z$ and $Z^2$), or something else? $\endgroup$
    – whuber
    Apr 10, 2014 at 20:26


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