Expected value of a distribution function

Question

Supposed that you have two random variables: $X$ and $Y$. Is $\Pr(X<Y)$ the same as $E_Y(F_X(Y))?$

Answer 1

For independent random variables, yes. In particular, \begin{align} \Pr\{X<Y\} & = \int_{-\infty}^{+\infty}\int_{-\infty}^{y} f_{XY}(x,y)dxdy \\ & =\int_{-\infty}^{+\infty} f_Y(y)\int_{-\infty}^{y}f_{X|Y}(x)dx dy\\ & =\int_{-\infty}^{+\infty} f_Y(y)F_{X|Y}(y) dy\\ & = \mathbb{E}[F_{X|Y}(Y)]. \end{align} Note that if $X$ and $Y$ are independent then $F_{X|Y}(x) = F_{X}(x)$.

Answer 2

The following assertion is correct (note the $\le$): $$ P(X\le Y) = E_Y[ F_X(Y) ] $$ because you can condition on $Y$: $$P(X\le Y)= E_Y( P(X\le Y\mid Y )) = E_Y(F_X(Y)). $$ Consider $Y=X=$ constant to see why the assertion is false with $<$ in place of $\le$.