Suppose that $f$ is the probability density function of a probability measure $P$ which is absolutely continuous with respect to Lebesgue measure.
Suppose X is a uniform random variable on the interval $[0, 1]$. Is it possible to characterize the distribution of the random variable $Y := f(X) X$?
One example: suppose that $P$ is the uniform law on $[0, 1/t]$, $t \ge 1$. Then $Y$ is $0$ with probability $1-1/t$ and otherwise it is $tZ$ where $Z$ is distributed according to $P$.