Correlation

The correlation between r.v.s $X$ and $Y$ is

$$\text{Corr}(X,Y)=\frac{\text{Cov}(X,Y)}{\sqrt{\text{Var}(X)\text{Var}(Y)}}=\frac{\text{Cov}(X,Y)}{\text{SD}(X)\cdot \text{SD}(Y)}$$

given that both variances are greater than zero.

Properties

Location-scale transformations do not affect correlations, as long as we don’t multiply the variables by zero. So, we have $\text{Corr}(aX+b,cY+d)=\text{Corr}(X,Y)$ for $a>0,c>0$.

This implies correlation does not depend on units of measurement.

It must be true that $-1\le \text{Corr}(X,Y)\le 1$. A perfect correlation is given by $|\text{Corr}(X,Y)|=1$ and signifies a completely linear relationship between $X$ and $Y$.