Let $X$ and $Y$ be random variables with density $f_X(x)=1_{[0,1]}(x)$ and $f_Y(x)=\frac{1}{2}1_{[0,2]}(x)$.
How do we couple these such that $X\leq Y$?
Attempt:
To couple these we must find new variables $X'$ and $Y'$ that are distributed identically as $X$ and $Y$ respectively and satisfy $X'\leq Y'$.
I thought we could start by taking $Y'=Y$ and then construct $X'$ such that $X'$ has the same distribution as $X$, i.e. $P(X'<x)=x$ for $x\in [0,1]$ and $P(X'<x)=1$ for $x>1$.
Can we then do $X'=Y$ if $Y<1$ and something else if $Y>1$? What would that be?