To show: If |Cor(X,Y)| = 1, then there exists a, b ∈ R s.t Y = bX + a. Any ideas or hints to proceed?

Basically, I've to prove that if the absolute value of correlation b/w two random variables is 1, then they should be linearly related.

So far,

$$ |cor(X, Y)| = 1 $$

$$ \frac{|Cov(X, Y)|}{|\sigma(X) \sigma(Y)|} = 1 $$

$$ |Cov(X, Y)| = |\sigma(X) \sigma(Y)| $$

How to proceed further?

  • 3
    $\begingroup$ Hint: when is the Cauchy-Schwarz inequality actually an equality? $\endgroup$ – Watson Nov 17 '16 at 15:37
  • 1
    $\begingroup$ Your statement is only almost sure: consider $X \sim N(0,1)$ and $Y=-X$ when $X$ is rational and $Y=+X$ when $X$ is irrational $\endgroup$ – Henry Nov 17 '16 at 15:40
  • $\begingroup$ @Watson: Wiki says the equality holds when they are "linearly related". But I've to show/prove this formally and I don't know how to proceed. $\endgroup$ – rakudo Nov 17 '16 at 15:47

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