Suppose $U$ and $V$ are random variables jointly uniformly distributed over the square with corners $(0,0), (1,0), (1,1)$ and $(0,1)$. I need to find the CDF and PDF of $X$ defined by $X=UV$. $$\mathbb{P}(UV\leq x) = \begin{cases}0 \text{ if x > 1 or x< 0}\\ x \text{ if 0}\leq x \leq 1 \end{cases}$$
For the PDF, I'd replace $x$ with $1$. Can someone please verify if the CDF and PDF are correct?
If $x\geq1$ then $P(UV\leq x)=1$ (not $0$).
If $0<x<1$ then:
$$P\left(UV\leq x\right)=\int_{0}^{1}\int_{0}^{1}\mathsf{1}_{\left(-\infty,x\right]}\left(uv\right)dudv=\int_{0}^{1}\int_{0}^{\min\left(1,\frac{x}{v}\right)}dudv=\int_{0}^{1}\min\left(1,\frac{x}{v}\right)dv=\cdots$$
I leave the rest to you (split up in $0<v\leq x$ and $x<v<1$).