## Definition 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 $-1 \leq \rho \leq 1$, where $\rho = \pm 1$ indicates a perfect linear relationship and $\rho = 0$ indicates no linear association. > [!example]- If $\text{Cov}(X,Y) = 6$, $\text{Var}(X) = 9$, and $\text{Var}(Y) = 16$, what is $\rho$? > $ \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 $Y$.