Equational Axioms for Expected Value Operators

Abstract

An equational axiomatisation of probability functions for one-dimensional event spaces in the language of signed meadows is expanded with conditional values. Conditional values constitute a so-called signed vector meadow. In the presence of a probability function, equational axioms are provided for expected value, variance, covariance, and correlation squared, each defined for conditional values.Finite support summation is introduced as a binding operator on meadows which simplifies formulating requirements on probability mass functions with finite support. Conditional values are related to probability mass functions and to random variables. The definitions are reconsidered in a finite dimensional setting.