Let $X$, $Y$ be independent, uniformly distributed random varibles on $[0,1]$, Calculate the joint density $(UV,U)$, where $U=\min(X,Y)$ and $V=\max(X,Y)$
$$P(VU\leq t, U\leq s)=P(V\leq \frac{t}{s}, U\leq s)=(\frac{1}{2}\frac{t}{s}s+\frac{1}{2}\frac{t}ss)1_{t\geq s^2}=t1_{t\geq s^2}$$, so the density would be this differentiated with respect to s and t, but it would be equal to 0… so I guess something's wrong with my calculation.
Best Answer
I will show you how to get the answer for $0<t<s<1$ and leave the other cases to you. In this case the probability is $\int_0^{1}\int_0^{\min \{s,\frac t v\}} dudv=\int_0^{1} \min \{s,\frac t v\} dv=\int_0^{t/s} sdv+\int_{t/s}^{1} \frac t v dv=t-t\log\, (\frac t s)$.