Table of Contents
Preface
Chapter I
Chapter II
I- Simulation
II- The Montรฉ 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 Eulerโs Formula and Abraham De Moivreโs 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 Bayesโ Problem
5- The Second Bayesโ 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 Merรฉ 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- Chebyshevโs 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 Pascalโs 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 โ The Gamblerโs Ruin Program
5- Absorbing Markov Chains โ The Rise and Fall of Stock Prices Program
Chapter IX
I- Introduction
II- Nomenclature
III- Historical Review
IV- Albert Einsteinโs 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