Master's Thesis — Robert Righter

Shawnee State University · MS in Mathematics

Thesis Advisor: Dr. John Whitaker

Recursive Computation of Compound Random Variables with a Finite-Mixture Count Distribution

Abstract

Compound random variables provide a natural framework for modeling aggregate quantities subject to two layers of randomness: the distribution of the summands and the distribution of the count variable. While the concept is straightforward, calculating the exact probability distribution of a compound random variable can be a significant computational challenge. Direct methods involving multiple convolutions are often cumbersome and inefficient. A more elegant and powerful approach is the use of recursive methods, which allow for the efficient calculation of the probability P(S_N = k) based on the probabilities of preceding values.

The primary contribution of this work is the extension and simplification of this recursive method to the case where the compounding distribution is a finite mixture of Poisson, Binomial, and Negative Binomial distributions. A direct application of the recursive formula to this case leads to a computationally intensive, nested problem, where the mixing weights of the distribution must be recalculated at each step. Additionally, a closed-form expression is derived that computes these recursive weights directly, thereby eliminating some of the nested recursion and simplifying the calculation.

📄 Read the PDF of the thesis · 📊 View the results of the computation implementation

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This repository contains the LaTeX source for my Master's Thesis. The primary source file is righter_thesis.tex.

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Prerequisites

Either of the following is sufficient:

1. A TeX distribution that provides latexmk and pdflatex
2. Docker (the build script will compile in a TeX container)

Platform	Recommended distribution
Ubuntu / Debian	sudo apt-get install texlive-full
macOS	brew install --cask mactex
Windows	MiKTeX

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Building the PDF

Run the provided build script from the root of the repository:

bash build.sh

The script will compile righter_thesis.tex using latexmk (preferred) or fall back to three sequential pdflatex passes when latexmk is not available. If no local TeX tools are installed but Docker is available, the script uses blang/latex:ctanfull automatically. The finished document is written to righter_thesis.pdf in the same directory.

To override the Docker image:

TEX_DOCKER_IMAGE=blang/latex:ctanfull bash build.sh

Manual build (latexmk)

latexmk -pdf righter_thesis.tex

Manual build (pdflatex only)

pdflatex righter_thesis.tex
pdflatex righter_thesis.tex   # second pass for cross-references
pdflatex righter_thesis.tex   # third pass for TOC / list of figures

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Implementation

The implementation/ directory contains a Python 3 implementation of the recursive method described in the thesis.

Requirements: Python 3 (no external dependencies required)

Running the implementation

python3 implementation/implementation.py

This will run all built-in tests (verifying the recursive computation against expected values from the thesis examples) followed by Monte Carlo simulation comparisons that validate the recursive results numerically.

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License

This work is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).