We have a random variable $X\sim Gamma(3,c)$, so that means $f(x)=\frac{c^3}{\Gamma(3)}x^2e^{-cx} ; \ x>0$, with $c$ being appropriately selected scale parameter.
We also have $P(U_2 \leq x)=x^2$ and $P(U_1\leq x)=x; \ 0\leq x\leq 1$. $\ \ X,U_1,U_2$ are independent.
Let's define $X_2=U_2X$ and $X_1=U_1U_2X=U_1X_2$.
How to calculate the survival function of the random variable $Z_3=X-X_2$ and find out how $Z_3$ is distributed (distribution function)?
Hint: While calculating, use conditioning on $X$.
Any help would be greatly appreciated.