4th Edition
Fundamentals of Probability With Stochastic Processes
"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
This one- or two-term calculus-based basic probability text is written for majors in mathematics, physical sciences, engineering, statistics, actuarial science, business and finance, operations research, and computer science. It presents probability in a natural way: through interesting and instructive examples and exercises that motivate the theory, definitions, theorems, and methodology. This book is mathematically rigorous and, at the same time, closely matches the historical development of probability. Whenever appropriate, historical remarks are included, and the 2096 examples and exercises have been carefully designed to arouse curiosity and hence encourage students to delve into the theory with enthusiasm.
New to the Fourth Edition:
- 538 new examples and exercises have been added, almost all of which are of applied nature in realistic contexts
- Self-quizzes at the end of each section and self-tests at the end of each chapter allow students to check their comprehension of the material
- An all-new Companion Website includes additional examples, complementary topics not covered in the previous editions, and applications for more in-depth studies, as well as a test bank and figure slides. It also includes complete solutions to all self-test and self-quiz problems
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 on applied probability, stochastic processes, and queuing theory.
- Axioms of Probability
- Combinatorial Methods
- Conditional Probability and Independence
- Distribution Functions and Discrete Random Variables
- Special Discrete Distributions
- Continuous Random Variables
- Special Continuous Distributions
- Bivariate Distributions
- Multivariate Distributions
- More Expectations and Variances
- Sums of Independent Random Variables and Limit Theorems
- Stochastic Processes
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
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
Reduction of Sample Space
The Multiplication Rule
Law of Total Probability
Bayes’ Formula
Independence
Chapter Summary
Review Problems
Self-Test on Chapter
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
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
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
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
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
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
Expected Values of Sums of Random Variables
Covariance
Correlation
Conditioning on Random Variables
Bivariate Normal Distribution
Chapter Summary
Review Problems
Self-Test on Chapter
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
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
Biography
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.
"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, University of Wisconsin
