Strength of a relationship
The correlation coefficient summarises the strength of the relationship between two variables.
Properties of correlation
For any random variables, \(X\) and \(Y\),
\[ -1 \;\le\; \Corr(X,Y) \;\le\; +1 \](Proved in full version)
The next result shows that the correlation coefficient between two variables can only be ±1 if the variables are linearly related.
Linear relationships and correlation
\[ \left|\Corr(X,Y)\right| = 1 \quad\text{if and only if} \quad Y = a + bX \quad\text{for some constants } a \text{ and } b. \](Proved in full version)