Probability: Modeling and Applications to Random Processes
This book takes an innovative approach to calculus-based probability theory, considering it within a framework for creating models of random phenomena. The author focuses on the synthesis of stochastic models concurrent with the development of distribution theory while also introducing the reader to basic statistical inference. In this way, the major stochastic processes are blended with coverage of probability laws, random variables, and distribution theory, equipping the reader to be a true problem solver and critical thinker.
Deliberately conversational in tone, Probability is written for students in junior- or senior-level probability courses majoring in mathematics, statistics, computer science, or engineering. The book offers a lucid and mathematicallysound introduction to how probability is used to model random behavior in the natural world. The text contains the following chapters:
* Sets and Functions
* Probability Laws I: Building on the Axioms
* Probability Laws II: Results of Conditioning
* Random Variables and Stochastic Processes
* Discrete Random Variables and Applications in Stochastic Processes
* Continuous Random Variables and Applications in Stochastic Processes
* Covariance and Correlation Among Random Variables
Included exercises cover a wealth of additional concepts, such as conditional independence, Simpson's paradox, acceptance sampling, geometric probability, simulation, exponential families of distributions, Jensen's inequality, and many non-standard probability distributions.
To the Student.
To the Instructor.
Chapter 1. Modeling.
1.1 Choice and Chance.
1.2 The Model Building Process.
1.3 Modeling in the Mathematical Sciences.
1.4 A First Look at a Probability Model: The Random Walk.
1.5 Brief Applications of Random Walks.
Chapter 2. Sets and Functions.
2.1 Operations with Sets.
2.3 The Probability Function and the Axioms of Probability.
2.4 Equally Likely Sample Spaces and Counting Rules.
Chapter 3. Probility Laws I: Building on the Axioms.
3.1 The Complement Rule.
3.2 The Addition Rule.
3.3 Extensions and Additional Results.
Chapter 4. Probility Laws II: Results of Conditioning.
4.1 Conditional Probability and the Multiplication Rule.
4.2 Independent Events.
4.3 The Theorem of Total Probabilities and Bayes' Rule.
4.4 Problems of Special Interest: Effortful Illustrations of the Probability Laws.
Chapter 5. Random Variables and Stochastic Processes.
5.1 Roles and Types of Random Variables.
5.3 Roles, Types, and Characteristics of Stochastic Processes.
Chapter 6. Discrete Random Variables and Applications in Stochastic Processes.
6.1 The Bernoulli and Binomial Models.
6.2 The Hypergeometric Model.
6.3 The Poisson Model.
6.4 The Geometric and Negative Binomial.
Chapter 7. Continuous Random Variables and Applications in Stochastic Processes.
7.1 The Continuous Uniform Model.
7.2 The Exponential Model.
7.3 The Gamma Model.
7.4 The Normal Model.
Chapter 8. Covariance and Correlation Among Random Variables.
8.1 Joint, Marginal and Conditional Distributions.
8.2 Covariance and Correlation.
8.3 Brief Examples and Illustrations in Stochastic Processes and Times Series.
- Presents a seamless introduction to the basic elements of probability together with applications from stochastic processes
- Places a primary focus on modeling so that students learn why various distributions and processes available are useful as probabilistic models of physical phenomena
- Briefly discusses methods of point estimation seamlessly throughout, rather than placing the topic in a separate chapter
- Prepares students for courses in statistical inference, stochastic processes, mathematical statistics, or other discipline-specific courses that require knowledge of probability and elementary stochastic processes
- Includes 395 exercises comprised of 916 separate parts that ask a total of over 1000 different questions across eight chapters; 90 additional conceptual questions finish out the chapters
"…a very pleasant and highly accessible textbook that perfectly meets the goal…[of making] probability theory accessible without sacrificing mathematical accuracy." (Mathematical Reviews, 2007h)
"This book more than lives up to its ambitious title…can hold its own against any comparable text." (MAA Reviews, January 30, 2007)
"This book is very useful for scientists and for students who study mathematics, statistics, economics and engineering." (Zentralblatt MATH, 1105,52)