References for stat 241

### Supplementary references for Statistics 241/541

• Sheldon Ross, A First Course in Probability
(Recommended as a clear source of good examples. Unfortunately, expectations are introduced in Chapter 7.)

• Frederick Mosteller, Fifty Challenging Problems in Probability with Solutions
(Most problems at the level of this course.)

• Jim Pitman, Probability
(A standard textbook at this level. I like it. Some students in previous years have liked it a lot.)

• J. L. Hodges and E. L. Lehmann, Basic Concepts of Probability and Statistics
Old text. Elementary treatment of some of the material in 241/541.

• William Feller, An Introduction to Probability Theory and Its Applications, Vol 1
(Widely known as Feller 1. A classic that is still a standard reference. Written by one of the grandmasters of probability theory.) Solid reading.

• Paul Hoel, Sidney Port, and Charles Stone, Introduction to Probability Theory
Old standard text. Slightly dated, in my opinion.

• Leo Breiman, Probability and Stochastic Processes: with a View Towards Applications
A good book for following up on some of the topics from 241/541. Chapter 6 contains a systematic account of Markov chain theory. Some parts require more mathematics than 241/541.

### Books about games and gambling

Many of my examples involve games of chance, one of the ancient forebears of modern probability theory.
• Edward Packel, The Mathematics of Games and Gambling
Fairly classical treatment. Interesting examples. Good background discussion.

• Richard Epstein, The Theory of Gambling and Statistical Logic
Slightly higher mathematical level than Epstein, but many parts within reach of 241/541. Some very interesting analyses.

• Abraham de Moivre, Doctrine of Chances: or, a Method of Calculating the Probabilities of Events in Play (Third edition 1756)
Contains solutions to several of the homework problems, but good luck to you if go looking. Historically, an immensely important book.

### Mathematical background

• Daniel Greene and Donald Knuth, Mathematicas for the Analysis of Algorithms
Computer Science orientation. Lots of combinatorics, manipulation of power series, and other pieces of classical mathematics.

• Arthur Engel, Elementary Mathematics from an algorithmic standpoint
Probability meets computers. Interesting discussion of some ideas related to 241/541. If you read German, his two-volume book on probability (cited in his references) makes a refreshing change from the old style.

Feller I is on my supplemental list, but the treatment is rather different from my approach. I place much emphasis on conditioning. Nevetheless, I have borrowed profitably from the book. I suspect most students would find it far too demanding as a text.

In general, I must note that the style of intro probability texts has changed a lot over the years. These days, much less emphasis on combinatorics, not as much formal math (no remainders in Taylor, if it is mentioned at all), more regard for the idea that an algorithm can be better than a messy closed form solution.

There are many books of the form "Probability and Statistics ...", in which the first part is devoted to probability. A book like Rice's "Mathematical Statistics and Data Analysis" comes to mind. Typically the treatment of probability is rather brisk, and maybe not quite what students can handle the first time around. We often use Rice for Stat 242.

Books like Olkin, Gleser, and Derman "Probability Models and Applications" cover similar material to 241, but the treatment of conditioning is too plodding. The conditional probability P(A|B) appears first in Section 3.3, introduced as a ratio; in 241, conditional probabilities appear in lecture 2, treated as a basic concept. They introduce conditional distributions and conditional expectations in Section 9.3; 241 Lecture 4 introduced conditional expectations. Their Chapter 12 contains the formal (classical) theory for Markov chains in discrete time; Lecture 3 in 241 treated a Markov chain problem.

I once used Chung "Elementary Probability Theory with Stochastic Processes" for 241, but I don't think the students liked it. It had some elegance.

I got mixed reviews when I used the Pitman text. Some students loved it; others found it too mathematical. I have adapted some topics from the book.

I feel that I must provide very detailed notes precisely because my approach to probability is different from most texts. Ultimately we all end up doing the same mathematical calculations, but the connection with reality differs in important ways. I learnt about the virtues of emphasizing conditioning from some old notes written by Terry Speed.