7. Probability and Statistics Soviet Essays - Sheynin, Oscar

Indeed, it can be shown by developing Markov’s ideas that both the law of large numbersand the Laplace limit theorem persist under most various dependences between the trials. Inparticular, this is true whatever is the dependence between close trials or individuals if only itweakens sufficiently rapidly with the increase in the distance from one of these to the otherone. Here, the coefficient of dispersion can take any value.Markov himself studied in great detail the simplest case of dependent trials forming (in histerminology) a chain. The population of letters in some literary work can illustrate such achain. Markov calculated the frequency of vowels in the sequences of 100,000 letters fromAksakov’s - (Childhood of Bagrov the Grandson) and of20,000 letters from {Pushkin’s} ! (Eugene Onegin). In both cases hediscovered a good conformity of his statistical observations with the hypothesis of a constantprobability for a randomly chosen letter to be a vowel under an additional condition that thisprobability appropriately lowers when the previous letter had already been a vowel. Becauseof this mutual repulsion of the vowels the coefficient of dispersion was considerably less thanunity. On the contrary, if the presence of a certain indicator, for example of an infectiousdisease in an individual, increases the probability that the same indication will appear in hisneighbor, then, if the probability of falling ill is constant for any individual, the coefficient ofdispersion must be larger than unity. Such reasoning could provide, as it seems to me, aninterpretation of many statistical series with supernormal but more or less constant dispersion,at least in the first approximation.Of course, until we have a general theory uniting all the totality of data relating to a certainfield of observations, as is the case with molecular physics, the choice between differentinterpretations of separate statistical series remains to some extent arbitrary. Suppose that wehave discovered that a frequency of some indicator has a normal dispersion; and, furthermore,that, when studying more or less considerable groups one after another, we have ascertainedthat even the deviations also obey the Gaussian normal law. We may conclude then that themean probability is the same for all the groups and, by varying the sizes of the groups, wemay also admit that the probability of the indicator is, in general, constant for all the objectsof the population; independence, however, is here not at all obligatory. If, for example, all theobjects are collected in groups of three in such a way that the number of objects havingindicator A in each of these is always odd (one or three), and that all four types of suchgroups (those having A in any of the three possible places or in all the places) are equallyprobable, then the same normality of the dispersion and of the distribution of the deviationswill take place as though all the individuals were independent and the probability of theirhaving indicator A were 1/2. Nevertheless, at the same time, in whatever random order ourgroups are arranged, at least one from among five consecutive individuals will obviouslypossess this indicator.It is clear now that all the attempts to provide an exhaustive definition of probability whenissuing from the properties of the corresponding statistical series are doomed to fail. Thehighest achievement of a statistical investigation is a statement that a certain simpletheoretical pattern conforms to the observational data whereas any other interpretation,compatible with the principles of the theory of probability, would be much more involved.[9] In connection with the just considered, and other similar points, the mathematicalproblem of generalizing the Laplace limit theorem and the allied study of the conditions forthe applicability of the law of random errors, or the Gaussian normal distribution, is ofspecial importance. Gauss was not satisfied by his own initial derivation of the law ofrandomness which he based on the rule of the arithmetic mean, and Markov and especiallyPoincaré subjected it to deep criticism 10 . At present, the Gauss law finds a more reliablejustification in admitting that the error, or, in general, a variable having this distribution, isformed by a large number of more or less independent variables. Thus, the normal