Let $X$ be the minimum and $Y$ the maximum of two independent, nonnegative random variables $S$ and $T$ with common continuous density $f$. Let $Z$ denote the indicator function of the event $(S < 3T)$. Are $(X,Z),(Y,Z),(X,Y)$ independent?
I am having a hard time doing this problem because the lack of density. After drawing the region defining $(s<3t)$ on a $(s,t)$ plane and visualizing it, I think $(X,Z)$ are independent which motivates me to show $P(XZ)=P(X)P(Z)$ which I really can't find a way to compute these probabilities without using density functions which were not given to me. Can you kindly give me some suggestion on how to do this problem?