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

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 $aZ+\log(c)$
