The Analysis of Selected Algorithms for the Stochastic Paradigm

Table of Contents
Preface
Chapter I
Chapter II
	I- Simulation
	II- The Monté Carlo Methods
	III- Random Numbers Generators
	IV- Matrices
		1- Definition
		2- Matrix Addition and Subtraction
		3- Scalar Multiplication
		4- Matrix Multiplication
		5- Matrix Transposition
		6- Symmetric Matrix; Skew-Symmetric Matrix
		7- Identity Matrix
		8- Null Matrix
	V- Numerical Methods
		1- Gauss-Jordan Method of Elimination to Solve a Linear System
		2- Gauss-Jordan Method with Pivoting
		3- Gauss-Jordan Method of Inversion
		4- Overdetermined Systems
	VI- Complex Numbers
		1- The Complex Number System
		2- Fundamental Operations with Complex Numbers
		3- Absolute Value
		4- Graphical Representation of Complex Numbers
		5- Vector Interpretation of Complex Numbers
		6- Leonhard Euler’s Formula and Abraham De Moivre’s Theorem
	VII- Conclusion
Chapter III
	I- Introduction
	II- The Theory
	III- Problems, Applications, and Algorithms
		1- The Cards Problem
		2- The First Box Problem
		3- The First Two Boxes P
		4- The First Bayes’ Problem
		5- The Second Bayes’ Problem
		6- The Second Box Problem
		7- The Coin Problem
		8- The Poker Problem
		9- The Fair Die Problem
		10- The Biased Die Problem
		11- The Books Problem
		12- The Chess Problem
		13- The Two Players Problem
		14- The Principle of Inclusion and Exclusion Problem
		15- The Birthday Problem
		16- The Second Two Boxes Problem
		17- The Bose-Einstein Problem
		18- The Fermi-Dirac Problem
		19- The Two Purses Problem
		20- The Bag Problem
		21- The Letters Problem
		22- The Yahtzee Problem
		23- The Strange Dice Problem
		24- The Equation Problem
		25- The De Moivre Problem
		26- The Huyghens Problem
		27- The Bernoulli Problem
		28- The De Meré Problem
		29- The Domino Problem
	IV- Conclusion
Chapter IV
	I- The Theory
		1- Random Variables
		2- Discrete Probability Distributions
		3- Distribution Functions for Random Variables
		4- Distribution Functions for Discrete Random Variables
		5- Continuous Random Variables
		6- Graphical Interpretations
		7- Joint Distributions
			7-1- Discrete Case
			7-2- Continuous Case
		8- Independent Random Variables
		9- Conditional Distributions
		10- Applications to Geometric Probability
	II- Problems, Applications, and Algorithms
		1- The Coin Algorithm
		2- The Second Coin Algorithm
		3- The Continuous Random Variable Algorithm
		4- The Joint Distribution Algorithm
Chapter V
	I- The Theory
		1- Definition of Mathematical Expectation
		2- Some Theorems on Expectation
		3- The Variance and Standard Deviation
		4- The Standardized Random Variables
		5- Moments
		6- Variance for Joint Distributions. Covariance
		7- Correlation Coefficient
		8- Chebyshev’s Inequality
		9- Law of Large Numbers
		10- Other Measures of Central Tendency
			10-1- Mode
			10-2- Median
		11- Skewness and Kurtosis
			11-1- Skewness
			11-2- Kurtosis
	II- Problems, Applications, and Algorithms
		1- The First Algorithm: Mathematical Expectation
		2- The Second Algorithm: Mathematical Expectation (Joint Distribution)
Chapter VI
	I- Introduction
	II- The Discrete Probability Distributions
		1- The Binomial Distribution
		2- The Geometric Distribution
		3- The Pascal’s or Negative Binomial Distribution
		4- The Hypergeometric Distribution
	III- The Continuous Probability Distributions
		1- The Normal Distribution
		2- The Standard Normal Distribution
		3- The Bivariate Normal Distribution
		4- The Gamma and Exponential Distributions
		5- The Chi-Squared Distribution
		6- The Cauchy Distribution
		7- The Laplace Distribution
		8- The Maxwell Distribution
		9- The Student t-Distribution
		10- The Fisher F-Distribution
	IV- Conclusion
Chapter VII
	I- Introduction
	II- The Theory
		1- Random Processes
			1-1- Definition
			1-2- Description of A Random Process
		2- Characterization of Random Processes
			2-1- Probabilistic Descriptions
			2-2- Mean, Correlation, and Covariance Functions
		3- Classification of Random Processes
	III- Problems, Applications, and Algorithms
		1- The Simple Random Walk Problem
		2- The Random Walk of a Particle Problem
		3- The Random Walk of a Drunkard Problem
Chapter VIII
	I- Introduction
	II- The Theory
		1- Definition of a Markov Chain
		2- The Initial Probability Distribution
		3- The Probability Vector
		4- The Probability of Passing from State i to State j in n Stages
		5- Regular Markov Chain
		6- Long-Term Behavior of a Regular Markov Chain
		7- Absorbing State; Absorbing Markov Chain
		8- The Fundamental Matrix of an Absorbing Markov Chain
		9- The Expected Number of Steps Before Absorption
		10- The Probability of Being Absorbed
		11- The Average Time Between Visits
	III- Problems, Applications, and Algorithms
		1- Markov Chains and Transition Matrices Pro
		2- Regular Markov Chains Program
		3- Absorbing Markov Chains Program
		4- Absorbing Markov Chains – The Gambler’s Ruin Program
		5- Absorbing Markov Chains – The Rise and Fall of Stock Prices Program
Chapter IX
	I- Introduction
	II- Nomenclature
	III- Historical Review
	IV- Albert Einstein’s Contribution
	V- The Purpose and the Advantages of the Present Work
	VI- The Complex Probability Paradigm
		VI-1- The Original Andrey Nikolaevich Kolmogorov System of Axioms
		VI-2- Adding the Imaginary Part M
		VI-3- The Purpose of Extending the Axioms
	VII- The New Paradigm and the Diffusion Equation
	VIII- The Evolution of Pc, DOK, Chf, and MChf
	IX- A Numerical Example
	X- Flowchart of the Complex Probability Paradigm
	XI- Simulation of the New Paradigm
		XI-1- The Paradigm Functions Analysis For t = 3000 seconds
			XI-1-1- The Complex Probability Cubes
		XI-2- The Paradigm Functions Analysis For t = 1000 seconds
		XI-3- The Paradigm Functions Analysis For t = 100 seconds
	XII- The New Paradigm and Entropy
	XIII- The Resultant Complex Random Vector Z
		XIII-1- The Resultant Complex Random Vector Z of a General Bernoulli Distribution
		XIII-2- The General Case: A Discrete Distribution with N Equiprobable Random Vectors
		XIII-3- The Resultant Complex Random Vector Z and The Law of Large Numbers
	XIV- The Complex Characteristics of the Probability Distributions
		XIV-1- The Expectation in C = R + M
			XIV-1-1- The General Probability Distribution Case
			XIV-1-2- The General Bernoulli Distribution Case
		XIV-2- The Variance in C
		XIV-3- A Numerical Example of a Bernoulli Distribution
	XV- Numerical Simulations
	XVI- Conclusion and Perspectives
	XVII- The Algorithms
Chapter X
Bibliography and References