Probability Bracket Notation: Probability Space, Conditional Expectation and Introductory Martingales

TL;DR

This paper extends Probability Bracket Notation (PBN), inspired by Dirac notation in Quantum Mechanics, to express probability spaces, conditional expectation, and martingales, aiming to simplify probability modeling and analysis.

Contribution

It introduces a novel PBN framework for probability spaces, conditional expectation, and martingales, demonstrating its potential to simplify complex probability problems without explicit measure theory.

Findings

PBN can represent probability spaces associated with R.V and S.P.

Conditional expectation properties are expressed and proved in PBN.

Martingales are formulated within the PBN framework.

Abstract

In this paper, we continue to explore the consistence and usability of Probability Bracket Notation (PBN) proposed in our previous articles. After a brief review of PBN with dimensional analysis, we investigate probability spaces in terms of PBN by introducing probability spaces associated with random variables (R.V) or associated with stochastic processes (S.P). Next, we express several important properties of conditional expectation (CE) and some their proofs in PBN. Then, we introduce martingales based on sequence of R.V or based on filtration in PBN. In the process, we see PBN can be used to investigate some probability problems, which otherwise might need explicit usage of Measure theory. Whenever applicable, we use dimensional analysis to validate our formulas and use graphs for visualization of concepts in PBN. We hope this study shows that PBN, stimulated by and adapted from Dirac notation in Quantum Mechanics (QM), may have the potential to be a useful tool in probability modeling, at least for those who are already familiar with Dirac notation in QM.