($t \in R$ and $f(t) \in [0,1]$). I want to keep $X$ and $Y$ general, but if it is easier, you can assume that they are continuous rv's (or even, Normal rv's), and say, $f(t) = \exp(-t^2)$.
I think it is intuitively right, since $\Pr\{Y<t \mid X\} < f(t)$ for any value of $X$, then I can safely drop the conditioning and get $Pr\{Y<t\} < f(t)$. But is it theoretically correct, or can you give a counter-example?
If it is correct, can we use this kind of reasoning to prove that $X$ and $Y$ are independent, i.e. if $\Pr\{Y<t \mid X\} < f(t)$, then $X$ and $Y$ are independent?
Thanks!