Perfect positive correlation
= Perfect positive correlation
{title2=$\rho(X,Y)=1$}
For square-integrable variables of positive <variance>, correlation one is equivalent to $X-\mathbb EX=c(Y-\mathbb EY)$ with $c>0$. The squared norm of the difference between the two standardized centered variables is $2(1-\rho)$, proving the equivalence. For uncentered variables the relationship can have a nonzero intercept; it need not be proportionality.