# References — Chapter ᚋ(M): The gallery of chance *Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#ref-ch-13, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.* [Guide for AI readers](index.md) · [The chapter](ch13.md) *Each entry: one plain sentence saying exactly what it is cited for, its status, then every source with a note saying what that source actually carries. "Own computation" / "Own reading" entries are the author's own work, not outside sources.* ### 13.1 {#r-13.1} For a random variable with a probability density, every exact singleton has probability 0, although some value is drawn. Probability zero is not impossibility: an event can have probability 0 though the set of outcomes it names is not empty. *Status: confirmed* - [MIT OpenCourseWare, Probability and Random Variables, Lecture 18](https://ocw.mit.edu/courses/18-440-probability-and-random-variables-spring-2014/d06be636fc6ff9365caf6ef166eb2b23_MIT18_440S14_Lecture18.pdf) — defines a continuous variable through its density and states explicitly that the probability of any single point is zero. - [Almost surely, Wikipedia](https://en.wikipedia.org/wiki/Almost_surely) — read as raw wikitext: “the set of outcomes on which the event does not occur has probability 0, even though the set might not be empty”; “the dart will almost never land on a diagonal … even though the set of points on the diagonals is not empty, and a point on a diagonal is no less possible than any other point.” ### 13.2 {#r-13.2} Kolmogorov’s axioms make probability a nonnegative countably additive measure with P(Ω)=1, so P(∅)=0. *Status: confirmed* - [Probability axioms, Wikipedia](https://en.wikipedia.org/wiki/Probability_axioms) — (§§ First, Second and Third axiom): states the three Kolmogorov axioms — “The probability of an event is a non-negative real number,” “P(Ω) = 1” and countable additivity — and states “P(∅) = 0. The empty set is interpreted as the event that ‘no outcome occurs’, which is impossible.” - [Probability theory, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Probability_theory) — gives the axioms as 0 ≤ P(A) ≤ 1, P(U) = 1, and additivity over pairwise mutually exclusive events. - [A. N. Kolmogorov, Foundations of the Theory of Probability, Chelsea, New York, 1950](https://archive.org/details/foundationsofthe00kolm) — (Ch. I, §1 “Axioms”): the English translation of the 1933 Grundbegriffe, the primary source for the axioms this claim names. ### 13.3 {#r-13.3} The variance of a constant is zero; independence implies zero covariance, but zero covariance does not imply independence. Under independence E[XY] = E[X]E[Y], so the covariance is 0; X uniform on [−1, 1] and X² are dependent with covariance 0. Deviations from the mean always sum to zero, which is why they are squared before averaging. *Status: confirmed* - [NIST/SEMATECH e-Handbook, Mean Vector and Covariance Matrix](https://www.itl.nist.gov/div898/handbook/pmc/section5/pmc541.htm) — gives variance and covariance as centered products; substituting a constant makes every deviation zero. - [MathWorld, Covariance](https://mathworld.wolfram.com/Covariance.html) — “while statistically independent variables are always uncorrelated, the converse is not necessarily true.” It gives no dependent zero-covariance example; the card’s X-against-X² example is this book’s own. - [Covariance, Wikipedia](https://en.wikipedia.org/wiki/Covariance) — read as raw wikitext: covariance is defined for variables “with finite second moments”; “If X and Y are independent random variables, then their covariance is zero. This follows because under independence, E[XY] = E[X] · E[Y].” “The converse, however, is not generally true. For example, let X be uniformly distributed in [−1, 1] and let Y = X². Clearly, X and Y are not independent,” yet their covariance is 0. - [James Jones, Measures of Variation, Math 113 lecture notes, Richland Community College](https://people.richland.edu/james/lecture/m113/variation.html) — “The problem is that this summation is always zero. So, the average deviation will always be zero.” and “So, to keep it from being zero, the deviation from the mean is squared and called the “squared deviation from the mean.”” ### 13.4 {#r-13.4} The Gaussian integral is √(2π), so the normal density has the factor 1/√(2π); √(2π), 1/e, 2/π and ln 2 are transcendental, e itself having been the first number proved transcendental, by Charles Hermite in 1873. Sources differ on the year Laplace evaluated the integral: 1774 (Lee’s York note, from Laplace’s Mémoire of that year) or 1782 (Normal distribution, Wikipedia); corrections welcome. *Status: confirmed* - [NIST Digital Library of Mathematical Functions §7.2, Definitions](https://dlmf.nist.gov/7.2) — defines erf and gives erf(∞) = 1, from which the √(2π) normalization follows. §7.2 is a definitions page and does not itself display the Gaussian integral; the P. M. Lee note beside it proves the value outright. - [Encyclopaedia of Mathematics, Lindemann–Weierstrass theorem](https://encyclopediaofmath.org/wiki/Lindemann%E2%80%93Weierstrass_theorem) — records the transcendence consequences for e, π and logarithms of nonunit algebraic numbers; the displayed consequences then follow algebraically. - [P. M. Lee, The Probability Integral (University of York note, typeset 2016, signed “P.M.L.”)](https://www.york.ac.uk/depts/maths/histstat/normal_history.pdf) — describes Laplace’s 1774 evaluation and credits the square-and-polar proof to Poisson. This six-page note is not Stigler’s The History of the Normal Distribution, under which title it was previously cited here. - [Transcendental number, Wikipedia](https://en.wikipedia.org/wiki/Transcendental_number) — read as raw HTML: “The first number to be proven transcendental without having been specifically constructed for the purpose of proving transcendental numbers’ existence was e, by Charles Hermite in 1873.” - [Normal distribution, Wikipedia](https://en.wikipedia.org/wiki/Normal_distribution) — gives 1782 (read as raw wikitext): “It was Laplace who first calculated the value of the integral ∫ e^(−t²) dt = √π in 1782, providing the normalization constant for the normal distribution.” ### 13.5 {#r-13.5} In the classical secretary problem the optimal cutoff fraction and success probability tend to 1/e as n grows; finite-n cutoffs are integers near n/e. Dennis V. Lindley’s 1961 article is widely acknowledged as the first solution. *Status: confirmed* - [Thomas S. Ferguson, Who Solved the Secretary Problem?](https://www2.math.upenn.edu/~ted/210F10/References/Secretary.pdf) — traces Gardner’s February 1960 related popular problem and Lindley’s 1961 classical solution, and gives the asymptotic 1/e result: “In Martin Gardner’s Mathematical Games column in the February 1960 issue of Scientific American, there appeared a simple problem that has come to be known today as the Secretary Problem.” - [Cambridge repository, Problems of optimal choice](https://api.repository.cam.ac.uk/server/api/core/bitstreams/42b7175b-f179-4976-b50c-cc218f86e68b/content) — confirms Lindley’s first published classical solution and discusses Gardner’s 1960 account. - [Dennis Lindley, Wikipedia](https://en.wikipedia.org/wiki/Dennis_Lindley) — read as raw wikitext: Dennis Victor Lindley (1923–2013); “It is widely acknowledged that in 1961, Lindley was the first to solve the Secretary problem in a scientific article.” ### 13.6 {#r-13.6} Buffon’s needle, with needle length equal to line spacing, crosses a line with probability 2/π; Buffon published the solved version in 1777. 2/π is also the average of |cos x| (equally |sin x|) over a period: a full-wave rectified sine averages 2/π of its peak. *Status: confirmed* - [MathWorld, Buffon’s Needle Problem](https://mathworld.wolfram.com/BuffonsNeedleProblem.html) — gives the 1777 solved version and probability 2/π. - [Dominic Klyve, How to Calculate π: Buffon’s Needle (Calculus version), TRIUMPHS, Digital Commons @ Ursinus College](https://digitalcommons.ursinus.edu/triumphs_calculus/7/) — the TRIUMPHS mini-Primary Source Project (Klyve, 2019) for Calculus I, built on Buffon’s own text on the needle problem; the landing page gives no date for Buffon’s essay (1777 is on the MathWorld line). - [Rectifier, Wikipedia](https://en.wikipedia.org/wiki/Rectifier) — average of a rectified sine is 2/π of its peak, V_av = 2·V_peak/π: “The average and RMS no-load output voltages of an ideal single-phase full-wave rectifier are:” ### 13.7 {#r-13.7} Under the usual 365-day model, 23 is the first group size whose shared-birthday probability exceeds one half; the same complement calculation underlies birthday attacks on hash functions. The first publication of the problem is Richard von Mises’s, of 1939; it is generally attributed before that to Harold Davenport, around 1927, who did not publish it at the time. *Status: confirmed* - [MathWorld, Birthday Problem](https://mathworld.wolfram.com/BirthdayProblem.html) — gives the complement product, the 23-person threshold and the 0.507297 shared-birthday probability. - [NIST Special Publication 800-2, Public-Key Cryptography](https://nvlpubs.nist.gov/nistpubs/Legacy/SP/nistspecialpublication800-2.pdf) — derives the birthday attack against hash functions from the same collision calculation. - [Birthday problem, Wikipedia](https://en.wikipedia.org/wiki/Birthday_problem) — read as raw HTML: “The problem is generally attributed to Harold Davenport in about 1927, though he did not publish it at the time”, and “The first publication of a version of the birthday problem was by Richard von Mises in 1939”, the paper being Über Aufteilungs- und Besetzungswahrscheinlichkeiten, Revue de la faculté des sciences de l’Université d’Istanbul 4, 145–163, 1939. ### 13.8 {#r-13.8} In the standard three-door Monty Hall setup, switching wins with probability 2/3; Selvin published the problem in 1975 and Marilyn vos Savant popularized it in 1990. *Status: judgement* - [Monty Hall problem, Wikipedia](https://en.wikipedia.org/wiki/Monty_Hall_problem) — “After the problem appeared in Parade, approximately 10,000 readers, including nearly 1,000 with PhDs, wrote to the magazine, most of them calling Savant wrong”; it names the 1991 New York Times front-page story as its source, and cites Steve Selvin’s letter to The American Statistician 29 (1), February 1975 (doi:10.1080/00031305.1975.10479121), which first posed the problem. Editorial note: The letter figures are vos Savant’s reported estimates, not an archival count; the chapter now says so. ### 13.9 {#r-13.9} The expected value of a fair six-sided die is 7/2 and need not be an attainable face value. *Status: confirmed* - [Penn State STAT 414, Mathematical Expectation](https://online.stat.psu.edu/stat414/Lesson08) — computes the fair-die expectation as 3.5 and explains that it is a theoretical average, not a realized face. ### 13.10 {#r-13.10} Shannon’s 1948 theory defines logarithmic information and entropy; a certain event contributes zero, and Shannon explicitly connects the form to entropy in statistical mechanics. *Status: confirmed* - [Claude E. Shannon, A Mathematical Theory of Communication](https://people.math.harvard.edu/~ctm/home/text/others/shannon/entropy/entropy.pdf) — primary paper defining logarithmic information and entropy, including the statistical-mechanics connection. ### 13.11 {#r-13.11} A fair binary outcome has entropy 1 bit, and Shannon credited John Tukey with the term bit. *Status: confirmed* - [Claude E. Shannon, A Mathematical Theory of Communication](https://people.math.harvard.edu/~ctm/home/text/others/shannon/entropy/entropy.pdf) — credits J. W. Tukey for “bit” and defines the binary unit; evaluating the entropy formula at 1/2 gives one bit. ### 13.12 {#r-13.12} Mutual information, the quantity C. E. Shannon introduced in A Mathematical Theory of Communication in 1948, is zero exactly for independent variables; KL divergence is nonnegative and zero exactly when the two distributions agree. Zero covariance, by contrast, does not imply independence. *Status: confirmed* - [Mutual information, Wikipedia](https://en.wikipedia.org/wiki/Mutual_information) — (§ Definition, § Relation to Kullback–Leibler divergence): “I(X;Y) = 0 if and only if X and Y are independent random variables,” and I(X;Y) “is equal to zero precisely when the joint distribution coincides with the product of the marginals.” - [Kullback–Leibler divergence, Wikipedia](https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence) — states non-negativity by Gibbs’ inequality, with equality exactly when the two distributions are equal almost everywhere; in the language of Bayesian inference, “it is the amount of information lost when Q is used to approximate P.” - [Kullback and Leibler, On Information and Sufficiency](https://doi.org/10.1214/aoms/1177729694) — DOI record for the 1951 primary paper introducing the information measure. - [C. E. Shannon, A Mathematical Theory of Communication, Bell System Technical Journal 27 (1948), 379–423](https://doi.org/10.1002/j.1538-7305.1948.tb01338.x) — the Crossref record gives the title, the journal, volume 27, July 1948 and pages 379–423, author C. E. Shannon. This is the 1948 the #mutinfo0 card names in its quick line and therefore in every popup. - [Covariance, Wikipedia](https://en.wikipedia.org/wiki/Covariance) — read as raw wikitext: covariance is defined for variables “with finite second moments”; “If X and Y are independent random variables, then their covariance is zero. This follows because under independence, E[XY] = E[X] · E[Y].” “The converse, however, is not generally true. For example, let X be uniformly distributed in [−1, 1] and let Y = X². Clearly, X and Y are not independent,” yet their covariance is 0. ### 13.13 {#r-13.13} The sample mean of identically distributed integrable variables is unbiased by linearity of expectation; independence is not needed for that mean identity. The first central moment E[X − μ] is 0. *Status: confirmed* - [MIT OpenCourseWare, Random Variables in a Nutshell](https://ocw.mit.edu/courses/2-086-numerical-computation-for-mechanical-engineers-fall-2014/84a1a919d238bbaea0bbae1ce3b8be3c_MIT2_086F14_Random_Var.pdf) — derives E[X̄]=μ by linearity, calls X̄ unbiased, and notes that independence was not used. - [Central moment, Wikipedia](https://en.wikipedia.org/wiki/Central_moment) — read as raw wikitext: “The first central moment μ1 is 0 (not to be confused with the first raw moment or the expected value μ).” ### 13.14 {#r-13.14} OLS residuals sum to zero when the fitted model contains an intercept. *Status: confirmed* - [NIST/SEMATECH e-Handbook, Least Squares](https://www.itl.nist.gov/div898/handbook/pmd/section1/pmd141.htm) — defines the intercept model and the least-squares objective. It does not itself state that the residuals sum to zero; that is the one-line consequence below. - **Own computation** (the author's own work) — Minimizing Σ(yᵢ − β₀ − β₁xᵢ)² over β₀ gives the normal equation −2Σ(yᵢ − β₀ − β₁xᵢ) = 0, that is Σ residuals = 0. The step needs the intercept: a fit without one carries no such constraint. ### 13.15 {#r-13.15} Fisher’s 1925 text describes one in twenty as a convenient significance convention, not a universal theorem. The 1925 first edition has the passage in its ch. III: P = .05, or 1 in 20, is about 1.96, “nearly 2,” taken as a convenient limit; “Deviations exceeding twice the standard deviation are thus formally regarded as significant.” *Status: confirmed* - [R. A. Fisher, Statistical Methods for Research Workers, 13th edition](https://archive.org/details/in.ernet.dli.2015.137901/page/n55/mode/2up) — (Ch. III, §12, p. 44): “The value for which P = ·05, or 1 in 20, is 1·96 or nearly 2; it is convenient to take this point as a limit in judging whether a deviation is to be considered significant or not. Deviations exceeding twice the standard deviation are thus formally regarded as significant.” - [MacTutor Extras, R. A. Fisher, Statistical Methods: Introduction](https://mathshistory.st-andrews.ac.uk/Extras/Fisher_Statistical_Methods/) — reproduces part of that book’s first chapter (The Scope of Statistics; General Method, Calculation of Statistics; The Qualifications of Satisfactory Statistics). The “one in twenty” passage is in its third chapter and is not on this page. - [R. A. Fisher, Statistical Methods for Research Workers (1925), ch. III, Classics in the History of Psychology (York University, Toronto)](https://psychclassics.yorku.ca/Fisher/Methods/chap3.htm) — the 1925 text: “The value for which P =.05, or 1 in 20, is 1.96 or nearly 2 ; it is convenient to take this point as a limit in judging whether a deviation is to be considered significant or not.” and “Deviations exceeding twice the standard deviation are thus formally regarded as significant.” ### 13.16 {#r-13.16} The standard-normal 97.5th percentile is conventionally rounded to 1.96 for a two-sided 95% interval. 1.96 is an approximation of the percentile point. *Status: confirmed* - [NIST/SEMATECH e-Handbook, Sample Sizes and Critical Values](https://www.itl.nist.gov/div898/handbook/prc/section2/prc222.htm) — explicitly gives the 0.975 critical value as 1.96 for a 95% interval. - [97.5th percentile point, Wikipedia](https://en.wikipedia.org/wiki/97.5th_percentile_point) — read as raw wikitext: “The approximate value of this number is 1.96, meaning that 95% of the area under a normal curve lies within approximately 1.96 standard deviations of the mean.” ### 13.17 {#r-13.17} Markov chains first appeared in Markov’s 1906 work on dependent variables; for the displayed matrix, π=(1/3, 2/3) is stationary and the ℓ¹ displacement is zero. Markov-chain Monte Carlo builds a chain whose equilibrium distribution is the target, and mixing time measures how fast a chain approaches its stationary distribution. *Status: confirmed* - [Dictionary of Scientific Biography, Andrei Andreyevich Markov](https://mathshistory.st-andrews.ac.uk/DSB/Markov.pdf) — states that these chains first appeared in Markov’s 1906 paper extending the law of large numbers to dependent variables. - **(untitled)** (the author's own work) — Direct calculation from the matrix displayed in the chapter: (1/3, 2/3)[[1/2, 1/2], [1/4, 3/4]]=(1/3, 2/3), so πP−π=(0, 0) and its ℓ¹ norm is 0. - [Markov chain Monte Carlo, Wikipedia](https://en.wikipedia.org/wiki/Markov_chain_Monte_Carlo) — read as raw wikitext: “Given a probability distribution, one can construct a Markov chain whose elements’ distribution approximates it, i.e. the Markov chain’s equilibrium distribution matches the target distribution.” - [Markov chain mixing time, Wikipedia](https://en.wikipedia.org/wiki/Markov_chain_mixing_time) — read as raw wikitext: “the mixing time of a Markov chain is the time until the Markov chain is “close” to its steady state distribution.” ### 13.18 {#r-13.18} Pólya proved recurrence of simple lattice random walk in dimensions 1 and 2 and transience above 2; p₃=0.3405373296... and has the displayed closed form. *Status: confirmed* - [MathWorld, Pólya’s Random Walk Constants](https://mathworld.wolfram.com/PolyasRandomWalkConstants.html) — gives Pólya’s 1921 result, p(3), and the Watson and Glasser–Zucker formulas. ### 13.19 {#r-13.19} The Pascal–Fermat correspondence of 1654 is conventionally treated as an early foundation of mathematical probability, not as the literal invention of every probabilistic idea. *Status: judgement* - [Dictionary of Scientific Biography, Pierre de Fermat](https://mathshistory.st-andrews.ac.uk/DSB/Fermat.pdf) — describes the 1654 Pascal–Fermat correspondence over the division of stakes and its foundational role. - [MacTutor, History overview](https://mathshistory.st-andrews.ac.uk/HistTopics/History_overview/) — places Pascal and Fermat at the conventional beginning of mathematical probability. Editorial note: The prose now says “conventional early landmark” and “helped lay foundations,” avoiding a false unique-invention claim. ### 13.20 {#r-13.20} Richard Price presented Bayes’s posthumous essay to the Royal Society in 1763; Laplace independently developed a related inverse-probability method in 1774. Bayes formulated a specific case of the theorem; Laplace reproduced and extended his results in 1774, apparently unaware of Bayes’s work. *Status: confirmed* - [Thomas Bayes and Richard Price, An Essay towards solving a Problem in the Doctrine of Chances](https://www.gtfp.cs.rhul.ac.uk/pulskamp/Bayes/bayesessay-rjp.pdf) — primary text records that Price communicated the essay and that it was read on 23 December 1763. - [MacTutor, Bayes and the development of inverse probability](https://mathshistory.st-andrews.ac.uk/Strick/bayes.pdf) — states that Laplace independently derived another form in 1774. - [Thomas Bayes, Wikipedia](https://en.wikipedia.org/wiki/Thomas_Bayes) — read as raw wikitext: Bayes is “known for formulating a specific case of the theorem that bears his name”; the essay “was read to the Royal Society in 1763 after Bayes’s death. Richard Price shepherded the work through this presentation and its publication.” - [Bayes’ theorem, Wikipedia](https://en.wikipedia.org/wiki/Bayes%27_theorem) — read as raw wikitext: Laplace “reproduced and extended Bayes’s results in 1774, apparently unaware of Bayes’s work, and summarized his results in Théorie analytique des probabilités (1812).” ### 13.21 {#r-13.21} The central limit theorem has early forms in de Moivre and Laplace, later rigorous general forms, and its present name from Pólya in 1920. De Moivre’s approximation to the binomial is of 1733; Gauss used the normal law to justify least squares in 1809; and Laplace proved and presented the central limit theorem in 1810, two years before the Théorie analytique des probabilités. *Status: judgement* - [Central limit theorem, Wikipedia](https://en.wikipedia.org/wiki/Central_limit_theorem#History) — (§ History): quotes Tijms on de Moivre’s 1733 normal approximation, Laplace’s rescue of it in the Théorie analytique des probabilités (1812) and Lyapunov’s general 1901 proof; and states that “The actual term ‘central limit theorem’ … was first used by George Pólya in 1920 in the title of a paper.” - [George Pólya, Über den zentralen Grenzwertsatz der Wahrscheinlichkeitsrechnung und das Momentenproblem, Mathematische Zeitschrift 8 (1920), 171–181](https://doi.org/10.1007/BF01206525) — the primary paper whose title introduced the name. - [MacTutor, George Pólya](https://mathshistory.st-andrews.ac.uk/Biographies/Polya/) — credits Pólya with the name central limit theorem in 1920. - [Normal distribution, Wikipedia (History)](https://en.wikipedia.org/wiki/Normal_distribution) — read as raw HTML: “In 1809, Carl Friedrich Gauss showed that the normal distribution provides a way to rationalize the method of least squares”, and “it was Laplace who in 1810 proved and presented to the academy the fundamental central limit theorem, which emphasized the theoretical importance of the normal distribution”. So the #normal card’s three dates — de Moivre 1733, Gauss 1809, Laplace 1810 — are each the right year for a different thing, and the 1812 this entry gives elsewhere is the Théorie analytique, a third. Editorial note: Because hypotheses and priority vary among versions, the card uses a compact lineage rather than assigning one unique theorem to one date. ### 13.22 {#r-13.22} Bernoulli’s Ars Conjectandi (1713) contains an early law of large numbers; strong-law forms assert almost-sure convergence. *Status: confirmed* - [MacTutor, Jacob Bernoulli](https://mathshistory.st-andrews.ac.uk/Biographies/Bernoulli_Jacob/) — identifies Ars Conjectandi (1713) and Bernoulli’s law-of-large-numbers result. - [Wikipedia, Law of large numbers](https://en.wikipedia.org/wiki/Law_of_large_numbers) — distinguishes the weak law (convergence in probability) from the strong law (almost-sure convergence), and dates Bernoulli’s proof to Ars Conjectandi, 1713. ### 13.23 {#r-13.23} The standard normal density has mean 0, variance 1 and total area 1; about 68.27% and 95.45% lie within one and two standard deviations. The integral of e^(−x²/2) gives the normalizing factor √(2π), so the density’s total area is 1. *Status: confirmed* - [NIST/SEMATECH e-Handbook, Normal Distribution](https://www.itl.nist.gov/div898/handbook/pmc/section5/pmc51.htm) — gives the density, parameters and areas 0.6827 and 0.9545. - [P. M. Lee, The Probability Integral (University of York note, typeset 2016)](https://www.york.ac.uk/depts/maths/histstat/normal_history.pdf) — defines φ(z) = (1/√(2π)) exp(−½z²) and shows “the integral of φ from −∞ to ∞ is 1, and hence φ is a probability density function.” ### 13.24 {#r-13.24} Current reference sources establish p₃’s numerical value and closed forms but do not supply a proof that p₃ is rational or irrational. *Status: judgement* - [MathWorld, Pólya’s Random Walk Constants](https://mathworld.wolfram.com/PolyasRandomWalkConstants.html) — gives the value, integral and gamma-function closed form without an arithmetic-status proof. - [Gaunt, Nadarajah and Pogány, On Pólya’s random walk constants](https://research.manchester.ac.uk/en/publications/on-p%C3%B3lyas-random-walk-constants/) — peer-reviewed 2024 Proceedings of the AMS article giving current closed forms for return probabilities, while making no rationality or irrationality claim for p₃. Editorial note: This is a cautious literature-survey judgement, not a proof from silence: the chapter says no proof appears in the current sources surveyed rather than asserting permanent unknowability. ### 13.25 {#r-13.25} The variance of a constant is zero; the deviations of a sample from its arithmetic mean sum to zero; and ordinary least squares residuals sum to zero when the fitted model contains an intercept. *Status: confirmed* - [Variance, Wikipedia](https://en.wikipedia.org/wiki/Variance) — § Basic properties states it outright — “The variance of a constant is zero,” with the displayed identity Var(a) = 0; § Propagation adds that “if a constant is added to all values of the variable, the variance is unchanged” and “If all values are scaled by a constant, the variance is scaled by the square of that constant,” which together give Var(aX + b) = a²Var(X). - [Arithmetic mean, Wikipedia](https://en.wikipedia.org/wiki/Arithmetic_mean) — § Motivating properties gives (x₁ − x̄) + ⋯ + (xₙ − x̄) = 0 and states that the mean “is the only number for which the residuals (deviations from the estimate) sum to zero.” - [Simple linear regression, Wikipedia](https://en.wikipedia.org/wiki/Simple_linear_regression) — § Numerical properties states “The sum of the residuals is zero if the model includes an intercept term,” and that the regression line passes through the point of sample means. Editorial note: Added by the V14 prose pass. The pages cited at 13.3 and 13.14 define variance, covariance and the least-squares objective, but none of them states these three consequences; 13.25 carries each one explicitly, and the three sentences that use them were meant to cite it alongside their original footnote. Checked on 17 September 2026: only one of the three did, and no card in the book cited this entry at all. The three cards and the two missing prose runs were attached on that date. ### 13.26 {#r-13.26} A finite Markov chain that is irreducible and aperiodic has a unique stationary distribution, and its k-step transition matrix converges to a matrix every row of which is that distribution, so the chain approaches it from any starting distribution. *Status: confirmed* - [Markov chain, Wikipedia](https://en.wikipedia.org/wiki/Markov_chain) — § Stationary distribution on finite state space states “If the Markov chain is additionally irreducible and aperiodic, then there is a unique stationary distribution π. In this case P^k converges to a rank-one matrix in which each row is the stationary distribution π,” and § Convergence speed to the stationary distribution gives the rate as the ratio of the second to the first eigenvalue. Editorial note: Added by the V14 prose pass to carry the chapter’s convergence sentence, which 13.17 does not: 13.17 covers Markov’s 1906 priority and the stationary computation only. The chapter’s matrix P = [[1/2, 1/2], [1/4, 3/4]] has every entry positive, so it is irreducible and aperiodic and the theorem applies; its second eigenvalue is 1/4, so the distance to π falls by three quarters a day. ### 13.27 {#r-13.27} The chapter epigraph is Laplace’s, from the Essai philosophique sur les probabilites of 1814, translated as A Philosophical Essay on Probabilities. *Status: confirmed* - [Pierre-Simon Laplace, Wikipedia](https://en.wikipedia.org/wiki/Pierre-Simon_Laplace) — read as raw HTML: “One sees in this essay that the theory of probabilities is basically only common sense reduced to a calculus. It makes one estimate accurately what right-minded people feel by a sort of instinct, often without being able to give a reason for it,” and the page dates the Essai philosophique sur les probabilités to 1814. Editorial note: Added 17 September 2026 to source the chapter epigraph, which carried no marker. The epigraph uses the older English rendering “at bottom nothing but common sense reduced to calculus”; this source gives “basically only common sense reduced to a calculus.” Two translations of one sentence, not two claims. ### 13.28 {#r-13.28} Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Shannon adopted Tukey’s word ‘bit’ in his 1948 paper; Glasser and Zucker gave the gamma-function closed form in 1977; Hermite proved e transcendental in 1873. Glasser and Zucker’s 1977 paper gives Watson’s simple-cubic result in closed form in gamma functions. *Status: confirmed* - [Bit, Wikipedia](https://en.wikipedia.org/wiki/Bit) — read raw: “Claude E. Shannon first used the word “bit” in his seminal 1948 paper “A Mathematical Theory of Communication.” He attributed its origin to John W. Tukey, who had written a Bell Labs memo on 9 January 1947 in which he contracted “binary digit” to simply “bit..”” - [Pólya’s Random Walk Constants, Wolfram MathWorld](https://mathworld.wolfram.com/PolyasRandomWalkConstants.html) — read raw: “Watson (1939), McCrea and Whipple (1940), Domb (1954), and Glasser and Zucker (1977) showed that.” - [Transcendental number, Wikipedia](https://en.wikipedia.org/wiki/Transcendental_number) — read raw: “The first number to be proven transcendental without having been specifically constructed for the purpose of proving transcendental numbers’ existence was e, by Charles Hermite in 1873..” - [M. L. Glasser and I. J. Zucker, Extended Watson integrals for the cubic lattices, PNAS 74 (1977), PubMed Central](https://pmc.ncbi.nlm.nih.gov/articles/PMC431006/) — abstract: “A new closed form expression for Watson’s result on the simple cubic lattice is given in terms of gamma functions.” Editorial note: Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim. ### 13.29 {#r-13.29} The nat is the unit of information or information entropy based on natural logarithms, as the shannon (the bit) is for base-2 logarithms; Shannon entropy written with a natural logarithm is a quantity in nats. *Status: confirmed* - [Natural unit of information, Wikipedia](https://en.wikipedia.org/wiki/Nat_(unit)) — read via the REST endpoint (revision 1376988607): “The natural unit of information (symbol: nat), sometimes also nit or nepit, is a unit of information or information entropy, based on natural logarithms and powers of e, rather than the powers of 2 and base 2 logarithms, which define the shannon.” and “When the Shannon entropy is written using a natural logarithm … it is implicitly giving a quantity with nat as the unit..”