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Since a few days I tried figure this out but I am stuck. Imagine there is an random variable $X$ which is distributed log-normally such that there is an normal distributed $Z \sim N(\mu,\sigma^2)$ such that $X = \exp{(\mu + \sigma Z)}$.
What I want to consider is a new random variable $Y = c \cdot X^a$. It is clear that $Y$ is log-normal as well since one can write $Y = c\cdot\exp{(a\mu)}\exp{(a\sigma Z)}$, but can I define $\mu' = a\mu + \ln{c}$ and $\sigma' = |a|\sigma$ for my the distribution of $Y$?

Also what is the correct pdf $f_Y(\mu,\sigma^2)$ for $Y$ assuming one knows $f_X(\mu,\sigma^2) = \frac{1}{x\sqrt{2\pi\sigma^2}}e^{-\frac{(\ln{x}-\mu)^2}{2\sigma^2}}$ ?

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  • $\begingroup$ Did you intend to say either $Z \sim N(\mu,\sigma^2)$ and $X=\exp(Z)$, or $Z \sim N(0,1)$ and $X=\exp(\mu +\sigma Z)$? $\endgroup$
    – Henry
    Sep 4, 2021 at 17:01

1 Answer 1

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Hint:

  • If $Z = \log(X)$ then $aZ+\log(c)=\log(cX^a)$
  • If $X$ has a log-normal distribution then $Z$ and $aZ+\log(c)$ have normal distributions
  • If you know the mean and variance of $Z$, you can find the mean and variance of $aZ+\log(c)$
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