# First Steps Towards an Imprecise Poisson Process

## TL;DR

This paper introduces imprecise Poisson processes by replacing the single rate parameter with an interval, developing a theoretical framework and practical methods for computing bounds on expectations of event counts.

## Contribution

It extends the classical Poisson process to an imprecise setting with rate intervals, providing new theoretical insights and computational methods.

## Key findings

- Defined imprecise Poisson processes with rate intervals
- Developed methods to compute lower and upper expectations
- Established a theoretical framework for imprecise stochastic processes

## Abstract

The Poisson process is the most elementary continuous-time stochastic process that models a stream of repeating events. It is uniquely characterised by a single parameter called the rate. Instead of a single value for this rate, we here consider a rate interval and let it characterise two nested sets of stochastic processes. We call these two sets of stochastic process imprecise Poisson processes, explain why this is justified, and study the corresponding lower and upper (conditional) expectations. Besides a general theoretical framework, we also provide practical methods to compute lower and upper (conditional) expectations of functions that depend on the number of events at a single point in time.

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Source: https://tomesphere.com/paper/1905.05734