# For an log-normally distributed $X$, what is the pdf of $Y = c \cdot X^a$?

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 $$f_X(\mu,\sigma^2) = \frac{1}{x\sqrt{2\pi\sigma^2}}e^{-\frac{(\ln{x}-\mu)^2}{2\sigma^2}}$$ ?

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

• 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 $$aZ+\log(c)$$