Modern Probability Theory and Its Applications

Written by: Emanuel Parzen (Stanford University)

I generally hesitate when adding textbooks that were published before I was born. However, this one is was digitized and still is being offered on the Texas A&M University website. His obituary describes this book as “one of the classical texts in probability theory.”

Modern Probability Theory and Its Applications was written by Emanuel Parzen and published in 1960. He was still an Associate Professor of Statistics at Stanford when it was written. However, he would become a much-loved member of the Texas A&M faculty. He has worked and published on signal detection theory and time series analysis, where he pioneered the use of kernel density estimation (also known as the Parzen window in his honor).

While students attempting Modern Probability Theory and Its Applications should have had at least one semester of college calculus, it is not strictly for mathematics students. Parzen recognized that in all fields, we are faced with the question “What is the probability that it will work?” rather than the simpler “Will it work?” He felt that a basic course in probability theory should be part of the curriculum for all scientists, engineers, mathematicians, statisticians and mathematics teachers. However, he thought a basic course in probability would be beneficial to students who plan on taking courses in statistics, statistical physics, industrial engineering, communication engineering, genetics, statistical psychology, and econometrics.

Students with a stronger calculus background and/or needing exposure to more advanced topics will find Chapters 7 through 10 useful.

Part of what I love about this textbook is all of the examples and exercises. Parzen included over 160 examples, 120 theoretical exercises and 480 exercises. And, yes, the book contains the answers to the odd-numbered exercises.

Buying This Book

With such an important scholarly work, there are numerous options to buying your own physical copy of Modern Probability Theory and Its Applications.

Amazon offers everything from a brand new re-release edition from 2013 to used, original copies from the 1960s.

The price can range from several hundred dollars to under $10, if you’re lucky. For the most part, when they’re available, you can pick up one of the older, used versions for under $20.

Table of Contents for Modern Probability Theory and Its Applications

1. PROBABILITY THEORY AS THE STUDY OF MATHEMATICAL MODELS OF RANDOM PHENOMENA
1 Probability theory as the study of random phenomena
2 Probability theory as the study of mathematical models of random phenomena
3 The sample description space of a random phenomenon
4 Events
5 The definition of probability as a function of events on a sample description space
6 Finite sample description spaces
7 Finite sample description spaces with equally likely descriptions
8 Notes on the literature of probability theory
2 BASIC PROBABILITY THEORY
1 Samples and n-tuples
2 Posing probability problems mathematically
3 The number of “successes” in a sample.
4 Conditional probability
5 Unordered and partitioned samples–occupancy problems
6 The probability of occurrence of a given number of events
3 INDEPENDENCE AND DEPENDENCE
1 Independent events and families of events
2 Independent trials
3 Independent Bernoulli trials
4 Dependent trials
5 Markov dependent Bernoulli trials
6 Markov chains
4 NUMERICAL-VALUED RANDOM PHENOMENA
1 The notion of a numerical-valued random phenomenon
2 Specifying the probability law of a numerical-valued random phenomenon
Appendix: The evaluation of integrals and sums
3 Distribution functions
4 Probability laws
5 The uniform probability law
6 The normal distribution and density functions
7 Numerical n-tuple valued random phenomena
5 MEAN AND VARIANCE OF A PROBABILITY LAW
1 The notion of an average
2 Expectation of a function with respect to a probability law
3 Moment-generating functions
4 Chebyshev’s inequality
5 The law of large numbers for independent repeated Bernoulli trials
6 More about expectation
6 NORMAL, POISSON, AND RELATED PROBABILITY LAWS
1 The importance of the normal probability law
2 The approximation of the binomial probability law by the normal and Poisson probability laws
3 The Poisson probability law
4 The exponential and gamma probability laws
5 Birth and death processes
7 RANDOM VARIABLES
1 The notion of a random variable
2 Describing a random variable

3 An example, treated from the point of view of numerical n-tuple valued random phenomena
4 The same example treated from the point of view of random variables
5 Jointly distributed random variables
6 Independent random variables
7 Random samples, randomly chosen points (geometrical probability), and random division of an interval
8 The probability law of a function of a random variable
9 The probability law of a function of random variables
10 The joint probability law of functions of random variables
11 Conditional probability of an event given a random variable. Conditional distributions.
8 EXPECTATION OF A RANDOM VARIABLE
1 Expectation, mean, and variance of a random variable
2 Expectations of jointly distributed random variables
3 Uncorrelated and independent random variables
4 Expectations of sums of random variables
5 The law of large numbers and the central limit theorem
6 The measurement signal-to-noise ratio of a random variable

7 Conditional expectation. Best linear prediction
9 SUMS OF INDEPENDENT RANDOM VARIABLES
1 The problem of addition of independent random variables
2 The characteristic function of a random variable
3 The characteristic function of a random variable specifies its probability law
4 Solution of the problem of the addition of independent random variables by the method of characteristic functions
5 Proofs of the inversion formulas for characteristic functions
10 SEQUENCES OF RANDOM VARIABLES
1 Modes of convergence of a sequence of random variables
2 The law of large numbers
3 Convergence in distribution of a sequence of random variables
Tables
Answers to Odd-Numbered Exercises
Index

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