Probabilistic Analysis of a Random Determinant

Master's Thesis , 2019 , 60 Pages , Grade: 8.5

Mathematics - Stochastics

We are interested in the behaviour of a determinant with i.i.d. random variates as its elements. A probabilistic analysis has been done for such determinants of orders 2 and 3. We have considered some of the well known distributions, namely, discrete uniform, Binomial Poisson, continuous uniform, standard normal, standard Cauchy and exponential. We are able to give fiducial limits for the determinant using Chebyshev’s inequality for all the distributions discussed in the text (except standard Cauchy distribution for which expectation does not exist). The main objective is to find the probability distribution of the determinant when its elements are from any of the distributions stated above. The desired distribution has been approximated using the method of transformation in general but when this method could not produce desired results we relied on empirical results based on simulation.

Inhaltsverzeichnis (Table of Contents)

• INTRODUCTION
• LITERATURE REVIEW
• Chapter 1 Probabilistic analysis of a determinant of order 2 and 3 for i.i.d. Binomial, Poisson and discrete uniform elements
  • 1.1 Fiducial limits of D
  • 1.2 Theoretical distribution for D of order 2 with discrete uniform elements for k = 2, 3
  • 1.3 RESULTS
• Chapter 2 Predicting a determinant of order 2 and 3 for i.i.d. U[0, 0] elements
  • 2.1 Fiducial limits of D
  • 2.2 Pdf of D of order 2
  • 2.3 Simulated results
  • 2.4 Discussion
  • 2.5 Results
• Chapter 3 On modelling the distribution of a determinant filled with i.i.d. exponential variates
  • 3.1 Fiducial limits of D
  • 3.2 Pdf of D
    • 3.2.1 Theoretical method
    • 3.2.2 Experimental method
  • 3.3 Discussion
  • 3.4 Results
• Chapter 4 On an Interesting Application of Johnson SB distribution in modelling the distribution of a determinant for standard normal and Cauchy variates as its elements
  • 4.1 Pdf of D
    • 4.1.1 Theoretical method
    • 4.1.2 Experimental method
  • 4.2 Discussion
  • 4.3 Results
• CONCLUSION
• FUTURE SCOPE OF WORK
• REFERENCES

Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)

This work aims to analyze the behavior of determinants with independent and identically distributed (i.i.d.) random variables as their elements. The main objective is to determine the probability distribution of the determinant when its elements are drawn from various distributions, including discrete uniform, Binomial, Poisson, continuous uniform, standard normal, standard Cauchy, and exponential. The study focuses on determinants of order 2 and 3. Key themes explored in the text include:

• Probabilistic analysis of random determinants.
• Fiducial limits of determinants.
• Distribution of determinants under different random variable distributions.
• Application of transformation methods and simulation techniques to approximate distributions.
• Use of Johnson SB distribution for modeling determinants with standard normal and Cauchy elements.

Zusammenfassung der Kapitel (Chapter Summaries)

• Chapter 1 investigates the probabilistic analysis of determinants of order 2 and 3 with i.i.d. Binomial, Poisson, and discrete uniform elements. It examines the fiducial limits of the determinant and explores the theoretical distribution for a determinant of order 2 with discrete uniform elements.
• Chapter 2 delves into the prediction of determinants of order 2 and 3 with i.i.d. continuous uniform elements (U[0, 0]). The chapter discusses the fiducial limits, derives the probability density function (pdf) of the determinant of order 2, and presents simulated results.
• Chapter 3 focuses on modeling the distribution of a determinant filled with i.i.d. exponential variates. It explores the fiducial limits of the determinant and attempts to determine the pdf using both theoretical and experimental methods.
• Chapter 4 explores the application of Johnson SB distribution in modeling the distribution of a determinant for standard normal and Cauchy variates as its elements. The chapter discusses the theoretical and experimental methods used to determine the pdf and presents results.

Schlüsselwörter (Keywords)

The work revolves around the probabilistic analysis of random determinants, focusing on distributions of determinants with i.i.d. elements drawn from various common distributions like Binomial, Poisson, discrete uniform, continuous uniform, standard normal, standard Cauchy, and exponential. The research explores the application of techniques like transformation methods and simulation to approximate distributions, and highlights the utility of the Johnson SB distribution for modeling determinants with standard normal and Cauchy elements.

• Course: Integrated MSc in Mathematics and Computing
• Grade: 8.5
• Authors: Nikita Saha (Author), Soubhik Chakraborty (Author)
• Publication Year: 2019
• Pages: 60
• Catalog Number: V478567
• ISBN (eBook): 9783668965089
• ISBN (Book): 9783668965096
• Language: English
• Tags: probabilistic, analysis, random, determinant