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

## 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

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

So:

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

$\blacksquare$