I have recently encountered a problem requiring dealing with two inverse distribution functions simultaneously.
For a cumulative distribution function $\Psi$, the function $\Psi^{-1}:(0,1)\rightarrow \mathbb{R}$ is defined by $\Psi^{-1}(t)=\inf\{ x\in\mathbb{R}:\Psi(x)>t\}$. Here is the original problem.
I have solved question (2-1). When I try to prove the inequality in (2-2), I need show that $E[F^{-1}(U)\cdot G^{-1}(U)]\geq E[X\cdot Y]$. However, I have no idea what $F^{-1}(U)\cdot G^{-1}(U)$ is like and do not know how to deal with it. I think question (2-1) may be useful. This is all I have done. Any advice will be helpful. Thanks a lot!