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With any number of random variables in excess of 1, the variables can be stacked into a random vector whose i th element is the i th random variable. Then the variances and covariances can be placed in a covariance matrix, in which the (i, j) element is the covariance between the i th random variable and the j th one.
Pearson's correlation coefficient is the covariance of the two variables divided by the product of their standard deviations. The form of the definition involves a "product moment", that is, the mean (the first moment about the origin) of the product of the mean-adjusted random variables; hence the modifier product-moment in the name.
Note that standard usage is to have matrices for the variances and covariances of vector random variables. Given these innovative definitions, the RV-coefficient is then just the correlation coefficient defined in the usual way. Suppose that X and Y are matrices of centered random vectors (column vectors) with covariance matrix given by
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Correlation does not imply causation; Correlation function; Cross-correlation matrix; Correlation function (astronomy) Correlation function (quantum field theory) Correlation function (statistical mechanics) Correlation ratio; Coskewness; Covariance; Covariance function; Covariance matrix; Covariance operator; Cramér's V; Cross-correlation ...
The magnitude of the covariance is the geometric mean of the variances that are in common for the two random variables. The correlation coefficient normalizes the covariance by dividing by the geometric mean of the total variances for the two random variables.
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The distance correlation is derived from a number of other quantities that are used in its specification, specifically: distance variance, distance standard deviation, and distance covariance. These quantities take the same roles as the ordinary moments with corresponding names in the specification of the Pearson product-moment correlation ...