If $P(XLet $X$ and $Y$ are two continuous random variable and $$P(X<Y)=P(X<g(Y)),$$ for some convex function $g$.
Is it true that $g$ will always be a linear function?

Let $X<0$ (a negative random variable) and $Y\ge 0$ (a non-negative random variable) and $g(y)=y^2$ convex. Then $$P(X<Y)=P(X<g(Y))=1$$ and $g$ is not linear. (Or $X \sim U[0,1], Y \sim U[2,3]$ and $g(y)=y^2$.) 
