= Solution
The inverse <transformation of a random variable> is $u=e^{-v}$, with absolute <derivative> $e^{-v}$. The condition $0<u<b$ becomes $v>-\log b$. The <change of variables for probability densities> gives
$$
\boxed{f_{V_n}(v)=\begin{cases}b^{-1}e^{-v},&v>-\log b,\\0,&v\leq-\log b.\end{cases}}
$$
Equivalently, $V_n+\log b$ has the <exponential distribution> of rate one. Applying a separate measurable <transformation of a random variable> to each independent $U_n$ preserves <independence of random variables>, so the $V_n$ are <independent and identically distributed>.
Back to article page
