Completely stuck on this problem.
Let X and Y be non-negative random variables with an arbitrary joint probability distribution function. Let $$I(x,y)=\begin{cases}1,\quad if \:X>x,\:Y>y\\0,\quad otherwise\end{cases}$$Show that $$\int_0^\infty \int_0^\infty I(x,y)dxdy=XY$$ It seems to me that $I(x,y)$ is an indicator variable (Bernoulli?) with probability of success $\mathbb P((X,Y)>(x,y))$. Then the integral is sort of summing over the real plane for all the ordered pairs that give $1$. Thus the integral should evaluate to the area of total region in $R^2$ where those $(x,y)$ are less than $X,Y$. But I have no idea if that's correct, or how to formally go about this, and honestly it feels like I'm floundering here...