Distribution of transformed lognormal random variable

Question

I was wondering:

if $X \tilde{} N(0,1)$ then $P=\exp(x)$ is lognormally distributed. However, what is the distribution of $P=c\exp(x)-d$ Will it still be lognormally distributed?

Thanks for your help!

Answer 1

You can try to use the transformation formula, i.e., for $b>0$,

$Y = aX+b$ where $X \sim \ln \mathcal{N}(0,1)$, thus $g^{-1}(Y)=\frac{1}{a}(Y-b)$, hence $$ f_Y(y) = f_X(g^{-1}(Y))|\partial/\partial yg^{-1}(Y)|=f_X(\frac{1}{a}(Y-b))|1/a|. $$ I.e., $Y$ is shifted log-normal r.v., with $\ln \mathcal{N}(\ln b, 1)$ and $y>b$.