Kovariansene

Kovariansene (covariances) are a statistical measure of how two random variables vary together. For random variables X and Y with finite means μX and μY, the covariance is Cov(X,Y) = E[(X − μX)(Y − μY)]. A positive covariance indicates that X and Y tend to move in the same direction, while a negative covariance indicates opposite movement. If X and Y are statistically independent, Cov(X,Y) = 0; however, zero covariance does not necessarily imply independence except in special cases such as jointly normal variables.

Covariance is symmetric: Cov(X,Y) = Cov(Y,X). If a and b are constants, Cov(aX+b, cY+d) = ac Cov(X,Y). Cov(X,X) = Var(X). For a random vector X = (X1, ..., Xk), the covariance matrix Σ has entries Σij = Cov(Xi, Xj). The diagonal elements are variances, and the matrix is always positive semi-definite, reflecting nonnegative quadratic forms.

In practice, population covariances are often unknown and must be estimated from data. The sample covariance between X and Y is Sxy = Σi (xi − x̄)(yi − ȳ) /(n − 1). More generally, the sample covariance matrix S has entries Sij = Σi (xi − x̄i)(xj − x̄j) /(n − 1).

Kovariansene relate closely to the correlation coefficient, defined as ρ(X,Y) = Cov(X,Y) /(σX σY), where σX and σY are standard deviations. The correlation lies in [−1, 1], and Cov is zero when variables are uncorrelated, though not necessarily independent.

Applications span finance (portfolio variance), multivariate statistics, and machine learning. In a multivariate normal distribution, the covariance matrix fully characterizes the dependence structure up to the means.