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Chapters
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Table of Contents
1.1 Experiments, Outcomes, and Events 1.2 Probability Definitions 1.3 Probability Axioms 1.4 Probability of Union of Events and Union Bound 1.5 Probability of Intersection of Events and Independence 1.6 Conditional Probability 1.7 Law of Total Probability 1.8 Bayes' Rule
Chapter 2: Discrete Probability Distributions 2.1 Discrete Random Variables 2.2 Probability Mass Function (PMF) 2.3 Cumulative Distribution Function (CDF) 2.4 Common Discrete Probability Distributions 2.5 MATLAB Implementation and Visualization 2.6 Solving Problems with Discrete Distributions
Chapter 3: Continuous Probability Distributions 3.1 Continuous Random Variables 3.2 Probability Density Function (PDF) 3.3 Cumulative Distribution Function (CDF) 3.4 Common Continuous Probability Distributions 3.5 MATLAB Implementation and Visualization 3.6 Solving Problems with Continuous Distributions
Chapter 4: Expectations and Moments 4.1 Mean, Variance, and Standard Deviation 4.2 Higher Moments 4.3 Moment Generating Function 4.4 MATLAB Exercises for Computing Expectations, Variances, and Moments
Chapter 5: The Law of Large Numbers 5.1 The Weak Law of Large Numbers 5.2 Convergence in Probability 5.3 MATLAB Demonstrations 5.4 Conclusion
Chapter 6: The Central Limit Theorem 6.1 The Central Limit Theorem 6.2 MATLAB Demonstrations 6.3 Conclusion
Chapter 7: Confidence Intervals 7.1 Basics of Confidence Intervals 7.2 Types of Errors and Significance Levels 7.3 Gaussian (Normal) Confidence Interval 7.4 t-Student Confidence Interval 7.5 Choosing between Gaussian and t-Student Confidence Intervals
Chapter 8: Regression and Correlation 8.1 Introduction to Linear Regression 8.2 Correlation Coefficients 8.3 Implementing Regression Analysis in MATLAB 8.4 Interpreting the Results 8.5 Conclusion
Chapter 9: Monte Carlo Simulations 9.1 Principles of Monte Carlo Simulations 9.2 MATLAB Examples for Practical Applications 9.3 Conclusion
Chapter 10: Maximum Likelihood Estimation 10.1 Introduction to Maximum Likelihood Estimation 10.2 MATLAB Examples 10.3 Conclusion
Chapter 11: Continuous Random Vectors 11.1 Joint Probability Density Functions 11.2 Expected Value Vector, Correlation Matrix and Covariance Matrix 11.3 Gaussian Random Vectors 11.4 Linear Transformations of Gaussian Random Vectors
Appendix A: MATLAB Primer for Beginners Appendix B: Monte Carlo and Estimating Integrals
Acknowledgments
As the journey of writing "Understanding Probability with MATLAB" comes to a close, my heart is filled with gratitude for those who have been the pillars of support, inspiration, and motivation throughout this process.
To my wife and children, words fall short to express my appreciation. Your unwavering support and understanding have been my sanctuary. Balancing the demands of writing with family life can be challenging, yet your patience and love made this journey not only possible but also incredibly rewarding. Your sacrifices did not go unnoticed, and this achievement is as much yours as it is mine.
I owe a debt of gratitude to my parents. Their relentless encouragement to pursue education and knowledge not only shaped the person I have become but also instilled in me the perseverance required to undertake this endeavor. Their belief in the value of hard work and continuous learning has been a constant source of inspiration. This book, in many ways, is a testament to the values they have imbued in me.
To my students, who have been a constant source of inspiration, thank you. Your curiosity, questions, and passion for learning have fueled my desire to share knowledge in a way that is both accessible and engaging. This book is a reflection of the energy and enthusiasm you bring to the classroom every day. It is my hope that it serves as a valuable resource in your pursuit of understanding probability, offering you the tools and confidence to explore the fascinating world of mathematics.
"Understanding Probability with MATLAB" is a culmination of collective support, love, and inspiration from each one of you. Thank you for being a part of this journey. May we continue to explore, learn, and grow together in the vast and intriguing universe of probability.
—Dr. Nir Regev