Bernoulli coupling by a shared uniform random variable
= Bernoulli coupling by a shared uniform random variable
{c}
{title2=$\mathbb P(B_r\ne B_s)=|r-s|$}
For $r,s\in[0,1]$ and a <random variable> $U$ with the <uniform distribution> on $[0,1]$, set $B_r=\mathbb1_{\{U\leq r\}}$ and $B_s=\mathbb1_{\{U\leq s\}}$. Each has the corresponding <Bernoulli distribution>, and $\mathbb P(B_r\ne B_s)=|r-s|$, because disagreement occurs precisely between the two thresholds. This is a <maximal coupling> of the two laws.