Suppose that $X$ and $Y$ are independent and identically distributed. The claim is that $P(X<Y)=P(X>Y)=1/2$. How do I prove this?
My attempt
Since they are IID $f_X=f_Y$. So $P(X<Y)=\int\int_{s<t} f_X(s)f_Y(t)dsdt=\int\int_{s<t} f(s)f(t)dsdt$
If this is correct, I just don't know how the above integral evaluates to $1/2$