Fundamentals of Probability

Saeed Ghahramani is Professor of Mathematics and Dean of the College of Arts and Sciences at Western New England University. He received his Ph.D. from the University of California at Berkeley in mathematics and is a recipient of teaching awards from Johns Hopkins University and Towson University. His research focuses in applied probability, stochastic processes, and queuing theory.

Axioms of Probability
Introduction
Sample Space and Events
Axioms of Probability
Basic Theorems
Continuity of Probability Function
Probabilities and Random Selection of Points from Intervals
What Is Simulation?
Chapter Summary
Review Problems
Self-Test on Chapter

Combinatorial Methods
Introduction
Counting Principle
Number of Subsets of a Set
Tree Diagrams
Permutations
Combinations
Stirling's Formula
Chapter Summary
Review Problems
Self-Test on Chapter

Conditional Probability and Independence
Conditional Probability
Reduction of Sample Space
The Multiplication Rule
Law of Total Probability
Bayes' Formula
Independence
Chapter Summary
Review Problems
Self-Test on Chapter

Distribution Functions and Discrete Random Variables
Random Variables
Distribution Functions
Discrete Random Variables
Expectations of Discrete Random Variables
Variances and Moments of Discrete Random Variables
Moments
Standardized Random Variables
Chapter Summary
Review Problems
Self-Test on Chapter

Special Discrete Distributions
Bernoulli and Binomial Random Variables
Expectations and Variances of Binomial Random Variables
Poisson Random Variable
Poisson as an Approximation to Binomial
Poisson Process
Other Discrete Random Variables
Geometric Random Variable
Negative Binomial Random Variable
Hypergeometric Random Variable
Chapter Summary
Review Problems
Self-Test on Chapter

Continuous Random Variables
Probability Density Functions
Density Function of a Function of a Random Variable
Expectations and Variances
Expectations of Continuous Random Variables
Variances of Continuous Random Variables
Chapter Summary
Review Problems
Self-Test on Chapter

Special Continuous Distributions
Uniform Random Variable
Normal Random Variable
Correction for Continuity
Exponential Random Variables
Gamma Distribution
Beta Distribution
Survival Analysis and Hazard Function
Chapter Summary
Review Problems
Self-Test on Chapter

Bivariate Distributions
Joint Distribution of Two Random Variables
Joint Probability Mass Functions
Joint Probability Density Functions
Independent Random Variables
Independence of Discrete Random Variables
Independence of Continuous Random Variables
Conditional Distributions
Conditional Distributions: Discrete Case
Conditional Distributions: Continuous Case
Transformations of Two Random Variables
Chapter Summary
Review Problems
Self-Test on Chapter

Multivariate Distributions
Joint Distribution of n > Random Variables
Joint Probability Mass Functions
Joint Probability Density Functions
Random Sample
Order Statistics
Multinomial Distributions
Chapter Summary
Review Problems
Self-Test on Chapter

More Expectations and Variances
Expected Values of Sums of Random Variables
Covariance
Correlation
Conditioning on Random Variables
Bivariate Normal Distribution
Chapter Summary
Review Problems
Self-Test on Chapter

Sums of Independent Random Variables and Limit Theorems
Moment-Generating Functions
Sums of Independent Random Variables
Markov and Chebyshev Inequalities
Chebyshev's Inequality and Sample Mean
Laws of Large Numbers
Central Limit Theorem
Chapter Summary
Review Problems
Self-Test on Chapter

Stochastic Processes
Introduction
More on Poisson Processes
What Is a Queuing System?
PASTA: Poisson Arrivals See Time Average
Markov Chains
Classifications of States of Markov Chains
Absorption Probability
Period
Steady-State Probabilities
Continuous-Time Markov Chains
Steady-State Probabilities
Birth and Death Processes
Chapter Summary
Review Problems
Self-Test on Chapter

»The 4th edition of Ghahramani's book is replete with intriguing historical notes, insightful comments, and well-selected examples/exercises that, together, capture much of the essence of probability. Along with its Companion Website, the book is suitable as a primary resource for a first course in probability. Moreover, it has sufficient material for a sequel course introducing stochastic processes and stochastic simulation.« — Nawaf Bou-Rabee, Associate Professor of Mathematics, Rutgers University Camden, USA

»This book is an excellent primer on probability, with an incisive exposition to stochastic processes included as well. The flow of the text aids its readability, and the book is indeed a treasure trove of set and solved problems. Every sub-topic within a chapter is supplemented by a comprehensive list of exercises, accompanied frequently by self-quizzes, while each chapter ends with a useful summary and another rich collection of review problems.« — Dalia Chakrabarty, Department of Mathematical Sciences, Loughborough University, UK

»This textbook provides a thorough and rigorous treatment of fundamental probability, including both discrete and continuous cases. The book's ample collection of exercises gives instructors and students a great deal of practice and tools to sharpen their understanding. Because the definitions, theorems, and examples are clearly labeled and easy to find, this book is not only a great course accompaniment, but an invaluable reference.« — Joshua Stangle, Assistant Professor of Mathematics, University of Wisconsin - Superior, USA