1 Oscar Sheynin History of Statistics Berlin, 2012 ISBN 978-3 ...

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1 Oscar Sheynin History of Statistics Berlin, 2012 ISBN 978-3 ...

observing an asteroid during a year and applying the Bayesian approach, heobtained EN ≈ n/p. He was not satisfied with this pseudo-answer andassumed now that p was unknown. Again applying the Bayesian approachand supposing that p took with equal probability all values within theinterval [0; 1], he derived instead EN = (M/m)n.He could have written this formula at once; in addition, it was possible torecall the Laplace problem of estimating the population of France by sampledata (§ 7.1-5). It is nevertheless interesting that Poincaré considered theunknown number of the minor planets as a random variable.5) Without mentioning Gauss (1816, § 5), he (pp. 192 – 194) derived themoments of the normal distribution and proved that the density functionwhose moments coincided with the respective moments of the normal lawwas normal. This proposition was due to Chebyshev (1887a), see alsoBernstein (1945/1964, p. 420).Poincaré applied his investigation to the theory of errors and nonrigorouslyproved the CLT: for errors of sensiblement the same order andconstituting une faible part of the total error, the resulting error followedsensiblement the Gauss law (p. 206).Also for proving the normality of the sum of errors Poincaré (pp. 206 –208, only in 1912) introduced characteristic functions which did notconform to their modern definition. Nevertheless, he was able to apply theFourier formulas for passing from them to densities and back. Thesefunctions weref(α) = Σp x e αx , f (α) = ∫φ(x)e αx dx. He noted that f(α) = 1 + αEx/1! +α 2 Ex 2 /2! + … (1; 2)6) Homogeneous Markov chains. Poincaré provided interesting examplesthat might be interpreted in the language of these chains and their ergodicproperties.a) He (p. 150) assumed that all the asteroids moved along one and thesame circular orbit, the ecliptic, and explained why they were uniformlyscattered across it. Denote the longitude of a certain minor planet by l = at +b where a and b are random and t is the time, and, by φ(a; b), the continuousjoint density function of a and b. Issuing from the expectationEe iml = ∫∫φ(a; b)e im(at + b) da db(which is the appropriate characteristic function in the modern sense),Poincaré not very clearly proved his proposition that resembled thecelebrated Weyl theorem (beginning of § 10.8.4). The place of a planet inspace is only known with a certain error, and the number of all possiblearrangements of the asteroids on the ecliptic might therefore be assumedfinite whereas the probabilities of the changes of these arrangements duringtime period [t; t + 1] do not depend on t. The uniform distribution of theasteroids might therefore be justified by the ergodic property ofhomogeneous Markov chains having a finite number of possible states.b) The game of roulette. A circle is alternately divided into a largenumber of congruent red and black sectors. A needle is whirled with forcealong the circumference of the circle, and, after having made a great numberof revolutions, stops in one of the sectors. Experience proves that the119

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