I understand the basic intuition behind scale families and location families, but do not know how to solve this proof.
Prove that the random variable $Y>0$ is from a scale family if and only if the random variable $X= ln Y$ is from a location family.
Any help or links to similar problems is much appreciated! :)
First assume $Y > 0$ is from a scale family. Then $Y = \sigma Z$ for some r.v. $Z$ and $\sigma > 0$. Then,
$$ F_X(x) = P(\ln Y \leq x) = P(\ln\sigma + \ln Z \leq x) = P(\ln Z \leq x - \ln \sigma) = P(\hat{Z} \leq x + \mu), $$ where $\hat{Z} = \ln Z$ and $\mu = -\ln \sigma \in \mathbb{R}$. Thus, there exists a r.v. $\hat{Z}$ such that $X = \hat{Z} + \mu$. In other words, $X$ is from a location family.
For the other direction, let $X = \ln Y$ be from a location family. Then, there exists a r.v. $\hat{Z}$ such that $X = \hat{Z} + \mu$ and we have $$ F_Y(x) = P(\exp (\ln Y) \leq x) = P(\exp(\hat{Z} + \mu) \leq x) = P(Z \leq \sigma x), $$ where $Z = \exp(\hat{Z})$ and $\sigma = \exp(-\mu) > 0$. Thus, there exsts a r.v. $Z$ such that $Y = \sigma Z$, i.e. $Y$ is from the scale family.