7. Probability and Statistics Soviet Essays - Sheynin, Oscar

examine in the first place the curves of distribution corresponding to established conditionswhich regrettably do not occur often. A specimen of a stationary distribution most oftenencountered in statistical physics is the simple law of geometric progression, or the linearexponential law of the distribution of energy among uniform elements with one degree offreedom. The latter corresponds to a given total amount of energy possessed by the wholepopulation of the elements under consideration. The same exponential law also regulates theprocess of natural decay of the atoms of radioactive substances.A similar problem also corresponds to the economic issue of the stationary distribution ofwealth among the individuals of a given society. The law discovered here by Pareto canillustrate a methodologically correct approach to constructing the theoretical curves withwhich the appropriate curves of economic statistics should be compared when searching foran explanation of the deviations from these curves in the peculiarities of the social structureand in the dynamics {in the moving forces and trends} in the given society.I shall not dwell on the fundamental difference in stating the problem of the distribution ofenergy depending on whether we assume the Boltzmann hypothesis of continuity or thePlanck hypothesis of discontinuity. I only note that the latter leads to issues in finitecombinatorial analysis whereas the mathematical problem corresponding to the formerconsists in determining the most probable distribution of probabilities of a positive variablewith a given expectation; under some general assumptions the distribution sought isexponential. When dropping the condition of positivity but additionally assigning the valueof the expectation of the square, we arrive at the exponential function of the second degree,i.e., to the Maxwellian law of distribution of velocities to which the Gauss normal lawcorresponds.Generalizing this result further, we find that, under the same overall assumptions, the mostprobable curve corresponding to given moments of the first k orders is expressed by anexponential function with a polynomial of the k-th degree as its exponent. Therefore, if itoccurs that in some cases the method of moments applied in statistics is not only a technicaltrick applied for calculation, but that the moments of several lower orders are indeedsubstantially constant, then an exponential curve with an exponent of the correspondingpower should be regarded as a typical distribution of such a statistical population.A general mathematical theory of stationary statistical curves does not exist yet. Theirdetermination in some cases, as in the just considered instance, is reduced to a problem in thecalculus of variations, and, in other cases, to functional and integral equations. It is natural toapply the latter method {?} in biology where some law of heredity and selection plays thepart of an iterative function or operator determining the transformation of the curve ofdistribution of one generation into the curve of the next one. Inversely, issuing from thestatistical distribution of consecutive generations, it is possible to seek the simplest iterativelaws compatible with the given dynamical process. In particular, in this way it becamepossible to establish that the Mendelian law of heredity is almost the only such elementarylaw that, when selection is lacking, realizes stationary conditions beginning already with thesecond generation.[12] Finally, turning over to correlation theory, it should be indicated first of all, that,excluding biological applications, most of its practical usage is based on misunderstanding.The desire to express all non-functional dependences through correlation is natural. However,no technical improvements replacing the hypothesis of normal correlation by any curvilinearcorrelation are attaining this goal since in probability theory the concept of correlation,according to its meaning, assumes stationarity which consists in that every variable involvedpossesses some fixed law of distribution.It is therefore senseless to consider, for example, the correlation between the amount of{paper} money in a country and the cost of a given product or the mean wholesale index. As