Pearson Correlation Coefficient

Definition

${\rho }_{X,Y}=\frac{\mathrm{cov}(X,Y)}{{\sigma }_X{\sigma }_Y}=\frac{E[(X-{\mu }_X)(Y-{\mu }_Y)]}{{\sigma }_X{\sigma }_Y}$

Facts

Let $E(Y|X)$ is linear in $X$, that is $E(Y|X)=a+bX$, then the Conditional Expectation and Conditional Variance will be $E(Y|X)={\mu }_2+\rho \frac{{\sigma }_2}{{\sigma }_1}(X-{\mu }_1)$ $E[Var(Y|X)]={\sigma }_2^2(1-{\rho }^2)$