Verification of Expectation Properties for Discrete Random Variables in HOL

Abstract

One of the most important concepts in probability theory is that of the expectation of a random variable, which basically summarizes the distribution of the random variable in a single number. In this paper, we develop the basic techniques for analyzing the expected values of discrete random variables in the HOL theorem prover. We first present a formalization of the expectation function for discrete random variables and based on this definition, the expectation properties of three commonly used discrete random variables are verified. Then, we utilize the definition of expectation in HOL to verify the linearity of expectation property, a useful characteristic to analyze the expected values of probabilistic systems involving multiple random variables. To demonstrate the usefulness of our approach, we verify the expected value of the Coupon Collector's problem within the HOL theorem prover.