Consider a continuous probability distribution $f$ and two random variables $X, Y$ both are greater than equal to $0$ and they are not identical random variables. Let's say one can show that $E[X]^k \geq E[Y]^k$ for all $k \geq 0$ and suppose their moment generating function exists.

Can I conclude that $X \geq Y$ almost surely? This is my intuition

I can easily get $Pr[X^k \geq Y^k] \neq 0$ otherwise we get the contradiction. Or at least I want to find the lower bound of $Pr[X^k \geq Y^k]$.

Any comment is welcome and thanks for your help.