Expectation of positive random vector?

Question

For $x$ positive random variable, I can calculate Its expectation by $$E(x) = \int^\infty_0 1-F_x(t) \,dt$$

Now, suppose I have positive random vector (x,y), is there any equivalent result? What would happen if I calculate $$ \int^\infty_0 \int^\infty_0 1-F_{x,y}(t1,t2)\,dt1\,dt2$$

Answer 1

By definition $E(X,Y)$ is the vector $(EX,EY)$. If $X$ and $Y$ are non-negative then $E(X,Y) =((\int_0^{\infty} [1-F_X(t)] dt,(\int_0^{\infty} [1-F_Y(t)] dt)$.