Probability Theory — Course Notes (McMullen, Harvard)

curtis-mcmullen‘s lecture notes for a probability course taught at harvard-university in 2011, typeset from DVI and last revised 29 March 2021. 98 pages. The source that completed this spoke’s founding cluster and triggered the split.

The first line states what they are: “These course notes accompany Feller, An Introduction to Probability Theory and Its Applications, Wiley, 1950.” They are a companion, not a replacement. The section numbering tracks Feller’s, gaps and all — chapters XII and XIII are simply missing, and the second half restarts at I and II for the continuous-density material of Feller’s second volume. Read them as one mathematician’s route through a classic rather than a self-contained text.

What it covers

Sample spaces and events, combinatorial analysis, random walks, combinations of events, conditional probability, the binomial and Poisson distributions, normal approximation, unlimited sequences of Bernoulli trials, random variables and expectation, the law of large numbers, integral-valued variables and generating functions, ruin problems, and then the continuous side: the exponential and uniform densities, and special densities with randomization. See probability-theory.

The teaching style

The notes open on “some sources and uses of randomness, and philosophical conundrums” — nine of them, from the flipped coin and Fermat’s interrupted game to quantum indeterminacy, drug trials, and randomness as a tool in graph theory and internet routing. Number six is the one that sets the tone: “Randomness as a model (in reality only one thing happens). Paradox: what if a coin keeps coming up heads?”

The worked examples are the notes’ distinguishing feature, and they are chosen to bite:

• The mailman paradox. A mailman delivers n letters at random; the chance nobody gets the right one is about 1/e = 37%. But for n = 2 there are only two arrangements, so the answer is plainly 50%. The resolution is that “at random” names two different sample spaces — all functions from letters to boxes (the porter) versus all bijections (the mailman) — and the notes use the contradiction to force the distinction. This is the pedagogic move the notes repeat: state a formula, break it, then build the sample space properly.
• Birthdays on Jupiter. A Jovian day is 9.925 hours and a Jovian year 11.859 earth years, so there are 10,467 possible birthdays; the back-of-envelope estimate k ≈ √(1.4N) puts the paradox threshold at about 121 people.
• The rule of seven. That 10·log 2 ≈ 7 is why money at k% doubles in roughly 70/k years rather than 100/k.
• Benford’s law, derived rather than asserted: a leading digit of 1 occupies [0, log 2] of log X mod 1, so about 30% of the interval. Once the Dow reaches 1,000 it must double to change its first digit; at 9,000 a 10% rise takes it back to one.

Two habits show through. Estimates come with the arithmetic that produced them (“it is good to know that log(2) = 0.693147… ≈ 0.7”), and structures get named in the language of the rest of mathematics — binomial coefficients are glossed by the transitive action of Sₙ on k-element subsets, which is not how a service course would put it.

Tier and limits

• T1. Primary teaching material published by its author on his own university page, written by a research mathematician teaching inside his discipline (curtis-mcmullen).
• Notes, not a book. Terse by construction, with no exercises of its own, no solutions, and Feller assumed alongside. Sections are uneven — some are dense derivations, others a list of examples. A reader without Feller gets the itinerary and not the terrain.
• Numbering gaps are the source’s, not a transcription error. Chapters XII–XIII are absent and the roman numerals restart mid-document; that is what the file contains.
• Fifteen years old and it does not matter. None of this material dates. What dates in a probability course is the applications, and the notes have few.
• Held at raw/mcmullen-probability-theory.pdf; arrived via Telegram with a Facebook click tracker appended to the URL.

Cross-spoke context

• ../quant-trading-wiki — quant-bible-mit-sloan teaches conditional probability, Bayes, expectation and variance as things to know for a job interview, and names MIT’s 18.600 as the course to take. This is that course’s material with the career removed. Ruin problems, covered here, are where the two subjects genuinely touch.
• ../machine-learning-wiki — information-theory-inference-learning-algorithms runs its own probability primer (chapters 2–3, 23) before using it for inference. Same material, opposite purpose: MacKay needs it to build a posterior, McMullen is teaching the subject.
• ../optimization-algorithms-wiki — random walks and Monte Carlo methods there are applied stochastics; the theory is here.

Related

probability-theory · mathematics · curtis-mcmullen · harvard-university · corral-vector-calculus · thibos-fourier-analysis · synthesis