Covariance measures how two random variables vary together, defined as the expected product of their deviations from their respective means, Cov(X,Y) = E[(X - E[X])(Y - E[Y])] = E[XY] - E[X]E[Y]; it generalizes variance (Cov(X,X) = Var(X)) and is the essential tool for computing the variance of a sum of dependent random variables, since variance is not linear. Correlation standardizes covariance by dividing by the product of the standard deviations, yielding a dimensionless quantity bounded between -1 and 1 (a consequence of the Cauchy-Schwarz inequality), and measures the degree of linear association between two variables. This theory belongs to probability theory's treatment of joint distributions of random variables, and it establishes key relationships such as bilinearity of covariance, independence implying zero covariance (but not the converse), and closed forms for the covariance structure of standard multivariate discrete distributions such as the multinomial, binomial, and hypergeometric.
Covariance measures how two random variables vary together, defined as the expected product of their deviations from their respective means, Cov(X,Y) = E[(X - E[X])(Y - E[Y])] = E[XY] - E[X]E[Y]; it …