An example satisfying $P(X > Y) > \frac 12$, but $E_X < E_Y$

Question

I've been working in this particular problem for a while and I'm still stuck with it. Please, some help on how to do this.

Let $(Ω, F, P)$ be a probability triple where $Ω = [0, 1]$ , $F$ is Borel $σ$-algebra on $[0, 1]$ and $P$ is Lebesgue measure.

Give an example of random variables $X$ and $Y$ defined on $(Ω, F, P)$ such that $P(X > Y) > \dfrac 12$, but $E_X < E_Y$.

Thanks!!!

Answer 1

Hint: You will need to define $X(\omega)$ and $Y(\omega)$ for every $\omega\in\Omega$. Try something simple for $X$ like $X(\omega)=1$ always. For $Y$, try $Y(\omega)=0$ most of the time, but otherwise $Y$ is very big. For best results you should visualize: consider drawing a graph of each random variable and play around with choices for $Y(\omega)$.