Hyperbolic valued random variables and conditional expectation

TL;DR

This paper introduces hyperbolic valued random variables, explores their expectation and moments, develops hyperbolic binomial and Poisson distributions, and defines conditional expectation, emphasizing variables that can take zero divisor values.

Contribution

It presents the first comprehensive framework for hyperbolic valued random variables, including distributions and conditional expectation, with analysis of zero divisor values.

Findings

Hyperbolic binomial and Poisson distributions are developed.

Expectation properties are analyzed using idempotent decomposition.

Conditional expectation for hyperbolic variables is defined and studied.

Abstract

In this paper, we introduce the concept of hyperbolic valued random variables, their expectation and moments. We develop the hyperbolic analogue of Binomial and Poisson distributions. We study some of the properties of expectation on the basis of decomposition of a hyperbolic number into idempotent components. Finally we define conditional expectation of hyperbolic valued random variable and study some of its basic properties. Our random variable can take values which are zero divisors and this is the important part of this study.