Given $X$ and $Y$ two random variables. $X\sim N(0, 1)$.
My intuition is that $\mathbb E_Y[F_X(Y)] = F_X[\mathbb E_Y[Y]]$, where $F_X$ is the CDF of the normal distribution. Is this a known result? Under what conditions would this hold? What tools can I use to determine whether my intuition is correct?