Varxi

VarXi is a notation used in probability theory to denote the variance of a random variable Xi. It is a notational convenience rather than a separate mathematical object, employed to emphasize the variance of a specific variable in a given context.

The variance VarXi is defined as the expected squared deviation from the mean: VarXi = E[(Xi − E[Xi])^2]. Equivalently, VarXi = E[Xi^2] − (E[Xi])^2, provided the first and second moments exist (i.e., E[Xi] and E[Xi^2] are finite). For a vector-valued variable, the concept generalizes to the covariance matrix.

Key properties include: Var(aXi + b) = a^2 VarXi for any constants a and b; if Xi and Y are independent, Var(X + Y) = Var(X) + Var(Y); the variance is always nonnegative and VarXi = 0 if and only if Xi is almost surely constant.

Notational notes: some texts write Var(Xi) or Var Xi with varying spacing or fonts, which can reflect different stylistic conventions. In certain contexts VarXi may appear as a shorthand for the variance of a time index sequence or a specific model variable, so consistency within a document is important to avoid confusion.

Applications include quantifying data dispersion, assessing estimation uncertainty, and measuring risk in finance through portfolio variance. A common example is when Xi follows a normal distribution Xi ~ N(μ, σ^2), in which case VarXi = σ^2. See also variance, standard deviation, covariance, and moments.