Correlation

def. Correlation (Correlation Coefficient). Correlation is Covariance, but standardized (unitless)

$$\begin{array}{rl}{\rho }_{XY}& =\text{Corr}(X,Y)\\ & =\text{Cov}(\frac{X-{\mu }_X}{{\sigma }_X},\frac{Y-{\mu }_Y}{{\sigma }_Y})\\ & =\frac{1}{{\sigma }_X{\sigma }_Y}\text{Cov}(X-{\mu }_X,Y-{\mu }_Y)\\ & =\frac{\text{Cov}(X,Y)}{{\sigma }_X{\sigma }_Y}\end{array}$$

thm. Correlation Identities.

• $-1\le \rho \le 1$
• ${\rho }_{XY}=1\iff \exists a,b\in \mathbb{R}\wedge a>0|Y=aX+b$
• ${\rho }_{XY}=-1\iff \exists a,b\in \mathbb{R}\wedge a<0|Y=aX+b$
• ${\rho }_{aX,bY}=\text{Corr}(X,Y)$ i.e. linearly invariant

For R.V. $X,Y$, the following means the Same Thing §

• $X,Y\text{ are uncorrelated}$
• $\text{Cov}(X,Y)=0$
• $\text{Var}(X+Y)=\text{Var(X)+Var(Y)}$
• $\mathbb{E}(X\cdot Y)=\mathbb{E}(X)\cdot \mathbb{E}(Y)$

Independence [$X\perp Y$] implies all of the above, but any of the above does not imply independence.