The problem is in the title (sorry if that's not kosher) but I have absolutely no clue how to get started. What would be a good approach to take with this one? I don't believe I have even seen a $\Bbb P(X<Y)$ before!
2026-04-29 10:20:02.1777458002
Find two random variables X and Y such that P(X<Y)=2/3
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You are asked to find random variables $X,Y$ such that: $$P\left(\{\omega\in\Omega\mid X(\omega)<Y(\omega)\}\right)=\frac23$$
Hint:
Let $Y$ only take values in $\{0\}$ and let $X$ only take values in $\{-1,1\}$.
Then $\{X<Y\}=\{X=-1\}$.