Tales of StatisticiansSiméon-Denis Poisson

21 June 1781 (Pitviers) - 25 Apr 1840 (Paris)Poisson was born to modestly situated parents, and owed his career to the new scientific institutions created by the Revolution, which systematically sought and advanced students of promise. That process took a while to get started. Poisson's father, a notary, thought Siméon too slow to follow in his footsteps, and sent him to Fontainebleau to study surgery. Siméon proved to be not notably adroit at such basic techniques as bloodletting, and the first patient he treated without supervision died. He returned home in dismay, aged fifteen. Such are the hazards of ranking possible professions by their social prestige level, rather than by their required aptitude level.

In the meantime, Siméon's father had risen to become the mayor of the town of Pitviers. He had bought a new house and subscribed to the leading periodicals of the day. One of these, the journal of the École Polytechnique, intrigued young Siméon, who found it easy to solve the mathematics problems in it. He was sent back to Fontainebleau, this time to study mathematics. Within two years he had qualified to enter the École Polytechnique. Laplace and Lagrange both took an interest in him, and Laplace remained a strong supporter in Poisson's later career. Under their tutelage, Poisson worked hard. He also found time for literature, and is said to have memorized the works of Molière, Corneille, and Racine. And he found time to read the mathematical classics, being especially attentive to d'Alembert and Euler. This brought him up to date professionally, and he never wasted time on discovering the already known; his original work was thus really original, and his first efforts were publishable. In his second and final year at the École, aged 18, Poisson produced a memoir on finite differences which attracted the attention of Legendre. He was launched on a career that quickly proved to be a major one. He became a versatile applied mathematician, astronomer, and physical scientist. "Poisson's constant" in electricity, and "Poisson's ratio" in elasticity, are among his monuments in the international vocabulary.

Like many of the founders of statistical method, such as Jacob Bernoulli, Poisson did important work in the calculus and the theory of functions, leaving in posterity's dictionary the terms "Poisson's integral" and "Poisson brackets" (in differential equations). His work has important implications for fluid mechanics, heat, geomagnetism, and electricity (it is the Poisson equation, and only the Poisson equation, that describes the electrical potential in terms of the distribution of charges). In all, he published over 300 mathematical papers during a placid and productive lifetime. His 1811 Treatise on Mechanics, written when he was thirty, became the standard work on that subject (it was republished in 1833), and his work in Fourier series was the basis of later advances by Dirichlet and Riemann. He sought to popularize the statistical theories of Laplace among practical men, in such areas as the theory of artillery; his 1829 Mémorial de l'Artillerie defined the dispersion of shots in a probabilistic setting, where the shots follow what is now called a bivariate Gaussian distribution. He also considered a theory of errors in which the error distribution is not necessarily Gaussian. This promising line of thought is still awaiting further investigation.

Now we come to his work in statistics. Poisson's Law of Large Numbers (1835), a generalization of Bernoulli and an advance on de Moivre, was the direct inspiration for Quetelet, and determined the direction of what is called the Continental school of statistics. In 1837, Poisson's Sorbonne lectures on probability and decision theory were published. They contained as one detail the germ of an idea which was later developed or rediscovered by others. This was the Poisson distribution, which predicts the pattern in which random events of very low probability occur in the course of a very large number of trials. Poisson's own application of this model was to the deliberations of juries, and did not arouse much interest, either then or later. Gauss, who was reputed to have been considering such questions also at about the same time, did not publish his conclusions (but then, Gauss

oftenfailed to publish his conclusions).The Poisson distribution was rediscovered in von Bortkiewicz's 1898 book, ironically entitled The Law of Small Numbers. It was shown to fit such empirical data sets as numbers of suicides and accidental deaths. The classic Poisson data set, extracted by von Bortkiewicz from official records, is the number of soldiers in the Prussian Army who died in a given year from the kick of a horse. We give details of this problem in the appropriate Lesson.

Another type of data which is amenable to Poisson description is the distribution of yeast cells in a suspension. This was the application of Gosset, who worked for the Guinness brewery firm, and who had arrived at the Poisson distribution without knowledge of Poisson or von Bortkiewicz. The matter was clarified by the 1935 report of Berkson and others, who found that while white blood cells and yeast cells do, as Gosset had noted, follow the Poisson distribution, red blood cells bump into and repel each other in the process of settling on the microscope slide. These collisions result in a slightly more uniform distribution of red cells than the Poisson model predicts. Another classic Poisson case is the 1910 experiment of Rutherford, Geiger, and Bateman on the decay of radioactive substances.

This one detail has somewhat eclipsed the rest of Poisson's work in the eyes of posterity. Small wonder: Poisson probability profiles occur wherever rare events of known

overallprobability happen randomly over a large set of time or space divisions, and where variation in one direction is more likely than variation in the other. Such situations turn out to be very common. Theories of queuing (how many tellers a bank needs to have in order to reduce waiting in line to a certain level) and similar questions are rooted in the Poisson distribution. It has come to be an important model for understanding events, and for detecting the presence of nonrandom factors in those events. Advice to those contemplating the purchase of a statistics textbook: The demographic sheep may be separated from the scientific goats by the fact that the former do not, and the latter do, include a chapter on the Poisson distribution.Spend your money wisely. Go with the goats.

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