Absolute Value of Pearson Correlation Coefficient is Less Than or Equal to 1

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Theorem

Let $X$ and $Y$ be random variables.

Let the variances of $X$ and $Y$ exist and be finite.


Then:

$\size {\map \rho {X, Y} } \le 1$

where $\map \rho {X, Y}$ denotes the Pearson correlation coefficient of $X$ and $Y$.


Proof

\(\displaystyle \paren {\map \rho {X, Y} }^2\) \(=\) \(\displaystyle \paren {\frac {\map {\operatorname {Cov} } {X, Y} } {\sqrt {\var X \var Y} } }^2\) Definition of Pearson Correlation Coefficient
\(\displaystyle \) \(=\) \(\displaystyle \frac {\paren {\map {\operatorname {Cov} } {X, Y} }^2} {\var X \var Y}\)
\(\displaystyle \) \(\le\) \(\displaystyle 1\) Square of Covariance is Less Than or Equal to Product of Variances

So:

$\size {\map \rho {X, Y} } \le 1$

$\blacksquare$

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