> [!NOTE] Definition (PCC of Discrete Real-Valued Random Variables) > Let $X, Y$ be [[Square-Integrable Discrete Real-Valued Random Variable|square-integrable discrete real-valued random variables]]. The (Pearson) correlation coefficient of $X$ and $Y$ is given by $\rho(X,Y)= \frac{\text{Cov}(X,Y)}{\sigma(X)\cdot \sigma(Y)}$where $\text{Cov}$ denotes [[Covariance of Square-Integrable Discrete Real-Valued Random Variables|covariance]] and $\sigma$ denotes [[Standard Deviation of Square-Integrable Discrete Real-Valued Random Variable|standard deviation]]. # Properties By [[Correlation Coefficient of Square-Integrable Discrete Real-Valued Random Variables is Invariant Under Linear Transformations of Variables]], for all $a,b,c,d\in \mathbb{R},$ $\rho(aX+b,cY+d)=\rho(X,Y).$ By [[Correlation Coefficient of Square-Integrable Discrete Real-Valued Random Variables is Symmetric]], $\rho(X,Y)=\rho(Y,X).$ By [[Correlation Coefficient of Square-Integrable Discrete Real-Valued Random Variables involving Negative]], $\rho(X,-Y)=-\rho(X,Y).$ By [[Absolute Value of Correlation Coefficient of Square-Integrable Discrete Real-Valued Random Variables is Bounded Above by 1]], $-1\leq\rho(X,Y)\leq 1.$