Introduction to Probability Theory and Stochastic Processes

This book presents an innovative approach to teaching probability theory and stochastic processes based on the binary expansion of the unit interval. Departing from standard pedagogy, it uses the binary expansion of the unit interval to explicitly construct an infinite sequence of independent random variables (of any given distribution) on a single probability space. This construction then provides the framework to understand the mathematical formulation of probability theory for its use in applications.

Comprehensive, astute, and practical, Introduction to Probability Theory and Stochastic Processes is a clear presentation of essential topics for those studying communications, control, machine learning, digital signal processing, computer networks, pattern recognition, image processing, and coding theory.

1 Coin Tossing 1

2 Countable Sample Spaces 61

3 Conditional Probability in Countable Sample Spaces 105

4 Uncountable Sample Spaces 151

5 Continuous Random Variables 213

6 Expectation 245

7 Modeling Random Phenomena 267

8 Functions of One Random Variables and Transforms 321

9 Functions of Two Random Variables 365

10 Two Functions of Two Random Variables 431

11 Conditional Probability for Continuous Random Variables 473

12 Random Vectors 549

13 Bernoulli, Geometric, and Poisson Processes 587

14 Brownian Motions and White Noise 645

15 Stationary Random Processes 703

16 Convergence of Random Variables 777

17 Statistics 839

18 Kalman Filter 905

Further Reading 933

Table of Common Distributions 935

References 941

Index 946