Elementary Probability (Stirzaker, 2nd edition)

David Stirzaker (Mathematical Institute and St John’s College, Oxford) — Cambridge University Press, 2nd edition, 2003, 538 pages. Ten chapters (0–9) in three parts, plus a review appendix of mathematical prerequisites and an appendix of solutions.

The spoke’s second account of probability-theory, and the first subject here to be covered twice by texts that overlap almost entirely. Until now the corpus’s two-account subject was vector-calculus, where the accounts disagree about foundations. These two agree about everything and differ in form: mcmullen-probability-theory is 98 pages of terse notes that assume Feller open beside them; this is a self-contained 538-page course with worked examples, exercises, problems and solutions.

What it covers

Three parts of three chapters each, with a new introductory chapter 0 in front.

• Part 1 — the basic ideas. Probability (sample spaces, addition rules, sequences of events), conditional probability and independence (including recurrence and difference equations), and counting (permutations, combinations, inclusion–exclusion, generating functions).
• Part 2 — discrete random variables. Distribution and expectation, then jointly distributed variables: independence, inequalities, conditional expectation, simple random walk, martingales, the law of averages, convergence. Then a chapter on generating functions — moments, sums of independent variables, joint and moment generating functions, regeneration, random walks.
• Part 3 — continuous random variables. Density and distribution, functions and simulation of random variables, ageing and survival, stochastic ordering; then jointly continuous variables (change of variables, order statistics, the Poisson process, two limit theorems).
• Chapter 9 — Markov chains. The Markov property, transition probabilities, first-passage times, stationary distributions, the long run, continuous-parameter chains, forward equations for Poisson and birth processes, and the Wiener process and diffusions.

Set against McMullen’s syllabus, the shared core is large — sample spaces, counting, conditional probability, random walks, expectation, generating functions, the law of large numbers, the continuous densities. What Stirzaker adds is everything that turns the subject into stochastic processes: martingales, the Poisson process, Markov chains, and Brownian motion.

What the second edition added, and why it matters here

The preface names four changes: the new chapter 0 on informal ideas about probability, a “Review and checklist” section closing every chapter, an elementary treatment of martingales, and an introduction to Brownian motion, diffusion and the Wiener process — the latter, in the author’s own framing, because it “underpinned much classical financial mathematics, such as the Black–Scholes formula for pricing options,” with optional stopping introduced alongside.

That is the first time this spoke’s corpus reaches the mathematics that ../quant-trading-wiki runs on, from the theory side. The trading spoke holds strategies and backtests; this holds the process models underneath them, taught as mathematics with the finance as an aside.

The pedagogy, which is where the two probability texts actually differ

Stirzaker states his aim as problem-solving, and gives a reason for it worth recording: “even at an elementary level, few problems are entirely routine. Successful problem solving requires flexibility and imagination.” So every chapter runs theory → many small examples → a block of worked examples with clustered exercises → end-of-chapter problems, with solutions to most exercises and many problems in an appendix. The worked examples carry names and are chosen partly for entertainment: Craps, Murphy’s Law, Poyla’s Urn, Gambler’s Ruin, the Genoese Lottery, the Ménages problem, Eddington’s Controversy, Dogfight, Ringing Birds.

One line in the preface is a direct contribution to this spoke’s rigour thread, because it is a pedagogical choice with a stated reason rather than a taste:

We adopt the now conventional formal definition of probability. This is not because of high principles, but merely because the alternative intuitive approach seems to lead more students into errors.

Formalism defended on error rates, not on rigour as a value. Set that beside keisler-elementary-calculus, which argues the opposite way about calculus — that the formal (ε-δ) route costs students the intuition for no gain — and the corpus has two authors making empirical-flavoured pedagogical claims in opposite directions, neither with data attached. Stirzaker’s is weaker evidence than Keisler’s, since Keisler at least cites a controlled experiment; “seems to lead more students into errors” is one teacher’s impression, stated as such.

Chapter 0 is the other notable choice: chance, models, symmetry, the long run, pay-offs, introspection, FAQs, history. It takes determinism seriously (Laplace at length, then quantum indeterminacy), and then does something the corpus should note — it closes the philosophical question rather than resolving it: “however long you ponder it, you will not produce a resolution of the problems and, second, none of this matters to our theory of probability.” McMullen’s notes open on nine philosophical conundrums too. Both probability texts here start with what randomness is, and both then decline to let the answer affect the mathematics.

Standing and limits

T1 — a Cambridge University Press textbook in its second edition, by an Oxford mathematician (the co-author, with Grimmett, of the standard Probability and Random Processes), carrying reviews from The Mathematical Gazette, the International Statistical Institute and Choice.

Three limits, stated plainly.

• It is in copyright and it is not open. © David Stirzaker 2003, CUP, “no reproduction of any part may take place without the written permission of Cambridge University Press.” The PDF sits on a third-party teaching site (ctanujit.org) whose right to host it is not established. This is the larsen-marx-mathematical-statistics situation again — accessible without being freely licensed — and the corpus’s free-by-licence pattern now has two exceptions, both of them commercially published statistics-and-probability texts.
• “Elementary” is relative. Set theory is assumed from the start, series and functions by part 2, calculus by part 3. The review appendix exists because the author knows this.
• It duplicates rather than extends the corpus’s coverage for its first six chapters. The genuinely new ground is martingales, Markov chains and Brownian motion, and those arrive at the end of a long book.

The URL as shared carried a Facebook click tracker (fbclid=…), stripped here — the same shape as the mcmullen-probability-theory link. Both probability sources in this spoke reached it through a social-media share.

Related

probability-theory · mcmullen-probability-theory · mathematical-statistics · keisler-elementary-calculus · larsen-marx-mathematical-statistics · david-stirzaker · university-of-oxford · mathematics