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A measure of the degree of association among a pair of variables.

The Pearson correlation between two random variables $$X$$ and $$Y$$ is defined as

$${\rm cor}(X,Y) = \frac{ E(XY) - E(X)E(Y) }{ \sqrt{ {\rm var}(X) {\rm var}(Y) } }$$

and is bounded between $$-1$$ (perfect negative linear relationship) and $$1$$ (perfect positive linear relationship). The numerator of $${\rm cor}(X,Y)$$ is known as the covariance between $$X$$ and $$Y$$.

If the Pearson correlation is $$0$$, we say the two variables are linearly independent.

Other definitions of correlations exist and can detect non-linear relationships. For example, Spearman's Rank Correlation' and Kendall's Rank Correlation