# What is direct and inverse variation

## The direct and the reverse (inverse) conclusion in heterograde theory

Introduction to Mathematical Statistics pp 193-266 | Cite as

### Summary

Before we begin our presentation, we believe we must again draw the reader's attention to the fact that the statistical aggregates with which the heterograde theory is confronted are incomparably more colorful and complicated than those which appear in the homograde theory. From the standpoint of the mathematician, the latter consist only of zeros and ones, which are mixed together in the proportion \ (\ frac {M} {N} ~ = ~ p ~ and ~ \ frac {NM} {N} = q \) (see above § 2, Chapter II), while the former can consist of any positive or negative whole or non-whole numbers, irrational numbers by no means excluded. From this it becomes evident that the propositions which express various general relationships between such totalities must be much more complicated on the one hand and much more indefinite on the other. If in the case of the homograde totalities one actually works with the quotients p and p'And all other statistical parameters that occur simply as functions of p, p′, N and n can represent, one has to do with the hetero-grade statistical totalities with an infinite series of statistical parameters: the series of mi, the μi, the product moments, the semi-invariants, the cumulants, etc. (see above Chapter II, § 5). And as long as no additional assumptions are made about the character of the distribution law of random variables, the exact binomial theorem (Chapter I, § 3) and the approximated exponential theorems by De Moivre-Laplace (Chapter I, § 4-6) and Poisson (Chapter I, § 12) only see Markoff adopting a generalized inequality (Chapter II, § 8), which, however, as we shall see later, allows a much greater range of variation for the probabilities in question and is significantly less advantageous in this respect . If one wants to specify the distribution law of the statistical totality more precisely, its variability has the effect that, depending on the concrete circumstances, a very large number of approximation formulas can be used and that, as it turns out, there is not a single one of them which in practice appears all cases would fit, because formulas that represent the law of distribution through an infinite series of statistical parameters can only be understood as "solutions" from the abstract mathematical and not from the practical-statistical point of view.

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### Special hint

This chapter is part of the Springer Book Archives digitization project with publications that have appeared since the publisher's inception in 1842. With this archive, the publisher provides sources for both historical and disciplinary research, which must be viewed in a historical context. This chapter is from a book that was published before 1945 and is therefore not advertised by the publisher in its political-ideological orientation typical of the time.

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