# Poisson processes and application to reliability theory and actuarial sciences

## Presentation

The Poisson process is widely used to model random "points" in time and space (e.g. claim arrivals in insurance, or failure times in reliability theory).

Furthermore, this model is widely used to construct number of other, more complex, random processes (e.g. compound processes to model the risk process in actuarial science).

The aim of this lecture is to introduce such models and study their fundamental properties from both probabilistic and statistical aspects, with application to reliability theory and actuarial science.

This lecture is divided into two parts :

Part I: Big picture

Introduction of the theoretical tools for :

• Homogeneous and inhomogeneous Poisson processes (definitions and basic properties).
• Inferential Statistics for homogeneous Poisson processes (estimation and tests for the process intensity).
• Simulation methods for Poisson processes.

Part II: Mini-Projects

Application and illustration on real and/or simulated data of different aspects of Poisson processes.

## Objectives

At the end of this module, the student should be able to :

• Know and understand the Poisson process theory fundamentals.
• Estimate the rate of a homogeneous Poisson process and construct confidence intervals and statistical tests for such rate (theoretically and in practice with the R Statistical Software).
• Model the recursive occurrences of the failures on a system, or the claim times in Insurance by Poisson processes.

## Form of assessment

The evaluation of outcome prior learning is made as a continuous training during the semester. According ot the teaching, the assessment will be different: as a written exam, an oral exam, a record, a written report, peers review...