VarXY

VarXY is a statistical construct used to describe the joint variability of two random variables X and Y. In common practice, VarXY refers to the covariance structure of the random vector (X,Y), often represented by the covariance matrix Σ, which captures Var(X), Var(Y), and Cov(X,Y).

Let μ be the mean vector μ = (E[X], E[Y]), and let Σ = Cov[(X,Y)]. If X and Y have finite variances, Σ is a 2x2 matrix: [Var(X) Cov(X,Y); Cov(Y,X) Var(Y)], with Cov(X,Y) = Cov(Y,X). The matrix is symmetric and positive semidefinite.

From samples {(xi, yi)}_{i=1}^n, the sample covariance matrix S estimates Σ: sxx = (1/(n-1)) Σ (xi − x̄)^2, syy = (1/(n-1)) Σ (yi − ȳ)^2, sxy = (1/(n-1)) Σ (xi − x̄)(yi − ȳ). Then S = [[sxx, sxy], [sxy, syy]].

Derived measures provide scalar summaries of the joint dispersion. For any real numbers a and b, Var(aX + bY) = a^2 Var(X) + b^2 Var(Y) + 2ab Cov(X,Y). The correlation ρ = Cov(X,Y)/sqrt(Var(X) Var(Y)). The generalized variance det(Σ) and the trace tr(Σ) are also used to assess overall spread in the X–Y plane.

Applications of VarXY appear in multivariate data analysis, regression diagnostics, and portfolio theory. In finance, VarXY helps quantify how two assets move together and contribute to portfolio risk. In science and engineering, it supports analysis of joint behavior between paired processes or measurements.

Notes on terminology: VarXY is not universally standardized. Some texts use Cov(X,Y) for the cross-covariance or refer to the 2x2 covariance matrix as VarXY; the term’s exact meaning can vary by discipline.