how can I prove the value of correlation coefficient $r$ ranges between $-1$ and $1$?

Question

What is the proof for the claim that the value of correlation coefficient $r$ ranges between $-1$ and $1$?

Answer 1

This is a consequence of Cauchy-Schwarz.

$$ \begin{align} \lvert \operatorname{cov}(X,Y)\rvert &= \lvert \mathbb{E}[(X-\mathbb{E}[X])(Y-\mathbb{E}[Y])] \rvert\\ &\leq \sqrt{\mathbb{E}[(X-\mathbb{E}[X])^2]}\sqrt{\mathbb{E}[(Y-\mathbb{E}[Y])^2]} = \sqrt{\operatorname{Var} X} \sqrt{\operatorname{Var} Y} \end{align}$$ and reorganizing gives the inequality $$ \begin{align} \left\lvert \frac{\operatorname{cov}(X,Y)}{\sqrt{\operatorname{Var} X} \sqrt{\operatorname{Var} Y}}\right\rvert &\leq 1 \end{align}$$

Answer 2

You can verify that the finite-mean random variables form a vector space on which covariance is an inner product. The correlation coefficient's range is then equivalent to the Cauchy-Schwarz inequality for that inner product.