Verification of probabilistic properties in HOL using the Cumulative Distribution Function

Traditionally, computer simulation techniques are used to perform probabilistic analysis. However, they provide inaccurate results and cannot handle large-scale problems due to their enormous CPU time requirements. To overcome these limitations, we propose to complement simulation based tools with higher-order-logic theorem proving so that an integrated approach can provide exact results for the critical sections of the analysis in the most efficient manner. In this paper, we illustrate the practical effectiveness of our idea by verifying numerous probabilistic properties associated with random variables in the HOL theorem prover. Our verification approach revolves around the fact that any probabilistic property associated with a random variable can be verified using the classical Cumulative Distribution Function (CDF) properties, if the CDF relation of that random variable is known. For illustration purposes, we also present the verification of a couple of probabilistic properties, which cannot be evaluated precisely by the existing simulation techniques, associated with the Continuous Uniform random variable in HOL.