VarYX

VarYX is a term used in statistics to denote the conditional variance of a random variable Y given another variable X. In this sense VarYX is equivalent to Var(Y|X). This quantity is a random variable that depends on X; it becomes a fixed number only when the value of X is known or when conditioning on a specific X.

Definition: Var(Y|X) = E[(Y − E[Y|X])^2 | X]. This expresses the expected squared deviation of Y from its conditional mean, given X. By the law of total variance, Var(Y) = E[Var(Y|X)] + Var(E[Y|X]).

Interpretation and usage: VarYX measures dispersion of Y within the subpopulations defined by X. It is central in heteroscedastic models, predictive uncertainty assessments, and Bayesian analysis where uncertainty in Y depends on X.

Examples: If Y conditional on X=x is distributed as Normal(mu_x, sigma_x^2), then Var(Y|X) = sigma_x^2, a function of x. The overall variance is Var(Y) = E[sigma_x^2] + Var(mu_x).

Notes: The notation VarYX is not universal; many texts simply write Var(Y|X). The concept is closely related to predictive variance, aleatoric uncertainty, and conditional expectation.

See also: Variance, Conditional expectation, Law of total variance, Regression analysis.