- No tags were found...

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 **of**homogeneous 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 number**of** revolutions, stops in one **of** the sectors. Experience proves that the119