Homogeneity and stability of statistical trials

Abstract

Homogeneity and stability, in their earlier historical setting, pertain in the main to independent binomial trials, where the probability of “success” in the ith trial is p _i . In the period of interest to us in this book, it was already well known that if p _i = p = const., n trials are considered, and X is the number of successes, then as n →∞, X/n → p in probability, that is, Pr(|X/n − p| ≥ є) → 0 for any є > 0 (Bernoulli’s Theorem). More recently, Poisson (1837b) had put forward the more general but heuristically motivated “rule” or “law,” which he called the Law of Large Numbers, that even without the assumption in probability p _i = const., $% MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaWGybGaai4laiaad6gacqGHsislceWGWbGbaebacaGGOaGaamOB % aiaacMcacqGHsgIRcaaIWaaaaa!3F7D! X/n - \bar p(n) \to 0$ in probability, where $% MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qaceWGWbGbaebacaGGOaGaamOBaiaacMcacqGH9aqpdaaeWaWdaeaa % peGaamiCa8aadaWgaaWcbaWdbiaadMgaa8aabeaaaeaapeGaamyAai % abg2da9iaaigdaa8aabaWdbiaad6gaa0GaeyyeIuoakiaac+cacaWG % Ubaaaa!4440! \bar p(n) = \sum\nolimits_{i = 1}^n {p_i } /n$. It was not rigorously demonstrated by Poisson and caused considerable confusion in regard to interpretation among the French probabilists, who therefore largely ignored it until the end of the century, although it had been proved (apparently unknown to them—surprisingly so in the instance of Bienaymé, in view of § 1.4) rigorously by Chebyshev (1846). We shall generally refer to it as Poisson’s Law of Large Numbers. Whereas Bernoulli’s Theorem in an obvious sense expresses stability of X/n, under the assumption of constancy or homogeneity of the p _i , Poisson’s law expresses, rather, the loss of variability of X/n.

History, regarded from the standpoint of unvarying causes, ... offers most useful lessons to mankind.

P. S. de Laplace