What distribution has a probability density function that is squared of the uniform distribution's pdf?

Question

What distribution has a probability density function that is squared of the uniform distribution's pdf $\dfrac{1}{b-a}$?

Answer 1

The pdf of $\operatorname{Unif}[a,b]$ is $\frac1{b-a}1_{[a,b]}$. Its square is $g=\left(\frac1{b-a}1_{[a,b]}\right)^2=\frac1{(b-a)^2}1_{[a,b]}$; moreover, $$\int_{\Bbb R}(b-a)^{-2}1_{[a,b]}\,dx=\frac1{b-a}$$ This indicates that $g$ is a pdf if and only if $b-a=1$, in which case it's the pdf of $\operatorname{Unif}[a,b]$.