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.
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, θ] 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
Objectives and Research Themes
This work aims to conduct a probabilistic analysis of determinants of order 2 and 3 where the elements are independent and identically distributed (i.i.d.) random variables. The research seeks to identify the distribution patterns of these determinants under various well-known statistical distributions, including discrete uniform, continuous uniform, Binomial, Poisson, exponential, standard normal, and standard Cauchy, providing fiducial limits and empirical approximations where theoretical methods are insufficient.
- Probabilistic behavior of determinants with i.i.d. elements
- Determination of fiducial limits using Chebyshev’s inequality
- Empirical approximation of probability density functions (pdf) through simulation
- Statistical modeling and fitting using the Johnson SB distribution
- Performance testing of distribution models via Kolmogorov-Smirnov and Anderson-Darling goodness-of-fit tests
Excerpt from the Book
INTRODUCTION
Is the distribution followed by a determinant identical to that of its elements or is it a different one? This particular question struck us and turned into our topic of interest at once.
Here, study has been done on the distribution assumed by a determinant of second and third order whose elements come from a particular distribution. We have considered some of the well known distributions namely—discrete uniform, continuous uniform, standard normal, standard Cauchy and exponential. A probabilistic analysis of the determinant has been done for each of these distributions in the different chapters of this text. Using the well known Chebyshev’s inequality in Probability, we are able to give the fiducial limits or confidence limits for the determinant. We have tried to approximate the distribution of the determinant theoretically in the case of discrete uniform and continuous uniform distributions and for the rest of the distributions the approximation has been done with the help of empirical results based on simulation.
Summary of Chapters
Chapter 1 Probabilistic analysis of a determinant of order 2 and 3 for i.i.d. Binomial, Poisson and discrete uniform elements: This chapter analyzes the distribution of determinants for discrete distributions, deriving fiducial limits via Chebyshev's inequality and providing theoretical distribution tables for specific cases.
Chapter 2 Predicting a determinant of order 2 and 3 for i.i.d. U[0, θ] elements: This section investigates determinants with continuous uniform elements, attempting transformation methods to find the pdf and supporting results with simulation data.
Chapter 3 On modelling the distribution of a determinant filled with i.i.d. exponential variates: This chapter focuses on determinants with exponential elements, where the Johnson SB model is identified as the best fit for the observed empirical 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: This final chapter applies the Johnson SB distribution to model determinants with continuous, non-bounded inputs like standard normal and Cauchy variables.
Keywords
Probabilistic Analysis, Random Determinant, i.i.d., Chebyshev’s Inequality, Fiducial Limits, Probability Density Function, Simulation, Johnson SB Distribution, Binomial Distribution, Poisson Distribution, Uniform Distribution, Exponential Distribution, Normal Distribution, Cauchy Distribution, Goodness of Fit
Frequently Asked Questions
What is the core focus of this research?
The research focuses on the probabilistic analysis of determinants of order 2 and 3 that consist of i.i.d. random variables, aiming to determine the resulting probability distribution of the determinant.
Which distributions are covered in the study?
The study examines determinants with elements following discrete uniform, continuous uniform, Binomial, Poisson, exponential, standard normal, and standard Cauchy distributions.
What is the primary methodology used?
The authors use a combination of theoretical analysis, primarily utilizing Chebyshev’s inequality for fiducial limits, and empirical analysis based on computer simulations when theoretical solutions are not feasible.
What role does the Johnson SB distribution play?
The Johnson SB distribution is utilized as the primary model to approximate the empirical distributions of the determinants, showing high goodness-of-fit results for various input types.
What is the goal of the simulations?
Simulations are conducted to generate empirical probabilities for the determinants, which are then used to test and validate the fit of different probability models using Kolmogorov-Smirnov and Anderson-Darling tests.
What are the key statistical indicators used to assess the models?
The study primarily utilizes Kolmogorov-Smirnov (K-S) statistics to rank and identify the best-fitting probability distributions for the determinant outcomes.
How is the fiducial limit derived for the determinants?
The fiducial limits are derived by calculating the mean and variance of the determinant of a random matrix and applying Chebyshev’s inequality to determine confidence intervals.
Are there limitations to the mathematical derivation of the distributions?
Yes, for several continuous distributions, the theoretical probability density functions of the determinants cannot be solved analytically, necessitating the reliance on empirical simulation results.
Does the order of the determinant change the analysis approach?
While the fundamental logic remains similar, the complexity of calculating the variance and expected values increases with the order of the determinant, requiring specialized derivations for orders 2 and 3.
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- Nikita Saha (Autor:in), Soubhik Chakraborty (Autor:in), 2019, Probabilistic Analysis of a Random Determinant, München, GRIN Verlag, https://www.grin.com/document/478567
