$$
\begin{align}
\rho(X,Y)&=1\begin{align}&\text{if and only if }\mathbb P(Y=\alpha X+\beta)=1\\&\text{ for some real }\alpha\text{ and }\beta\text{ with }\alpha>0\end{align}\\
\rho(X,Y)&=-1\begin{align}&\text{if and only if }\mathbb P(Y=\alpha X+\beta)=1\\&\text{ for some real }\alpha\text{ and }\beta\text{ with }\alpha<0.\end{align}\\
\end{align}
$$
