= Solution
For centered square-integrable variables with positive <variances>, the <correlation coefficient> is
$$
\rho(X,X')=\frac{\mathbb E[XX']}{\sqrt{\mathbb E[X^2]\mathbb E[(X')^2]}}.
$$
Put $v=\sqrt{\mathbb E[X^2]}$ and $v'=\sqrt{\mathbb E[(X')^2]}$. Then
$$
\mathbb E\left[\left(\frac Xv-\frac{X'}{v'}\right)^2\right]=2(1-\rho).
$$
Thus \b[$\boxed{\rho=1\iff X=(v/v')X'\text{ almost surely}}$], with positive proportionality factor. Conversely positive proportionality immediately gives correlation one. For variables with nonzero means, this criterion applies to their centered versions and yields an affine, not necessarily proportional, relationship. That distinction is essential for the positive stock prices below.
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