#
Fundamentals of Probability

With Stochastic Processes

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## Book Description

"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.

## Table of Contents

- 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

## Author(s)

### 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.

## Reviews

"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