Correlation Coefficient

Study guide: Exam P-1

The Correlation Coefficient (Pearson's $\rho$ ) is the standardized measure of linear association between two random variables, obtained by dividing the Covariance by the product of the standard deviations.

$\rho(X, Y) = \frac{\text{Cov}(X, Y)}{\sigma_X \cdot \sigma_Y}$

It satisfies $-1 \leq \rho \leq 1$
$\rho = \pm 1$ indicates a perfect linear relationship; $\rho = 0$ indicates no linear association

Computing the Correlation Coefficient from Variances If $\text{Cov}(X,Y) = 6$ , $\text{Var}(X) = 9$ , and $\text{Var}(Y) = 16$ , what is $\rho$ ?

Answer $\rho = \frac{6}{\sqrt{9} \cdot \sqrt{16}} = \frac{6}{3 \times 4} = \frac{6}{12} = 0.5$ This indicates a moderate positive linear relationship between $X$ and $Y$ .