Wahrscheinlichkeitsrechnung. Vol. II. Mathematische Statistik, Mathematische Grundlagen der Lebensversicherung
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... echnung. Vol. II. Mathematische Statistik, Mathematische Grundlagen der Lebensversicherung. By EMANUEL CZUBER. Second, revised and enlarged, edition. Pp. x+470. 8vo. 1910. (Teubner, Leipzig.) Professor Czuber's first edition, published in 1903, was in only one volume. By reserving for this second volume some of the principal applications of the fundamental theorems dealt with in the first, he is now able to introduce a considerable number of additions. Almost all the changes which Professor Czuber has made are by way of addition, and he has retained without alteration almost the whole text of the first edition. The most considerable of these enlargements are directed to a fuller treatment of the subject of invalidity. There are new sections on invalidity statistics, on invalidity insurance, and on State insurance. The mathematical theory of mortality statistics and of life insurance, the subject of invalidity being introduced as matter subsidiary to this, occupies the greater part of the volume now under review. So far as I can judge, these topics have been admirably treated, and the very numerous references to other writers ought to render it exceedingly valuable to students of the technical detail of this subject. But with technical detail it is mainly concerned, and there is not a great deal of much interest to mathematicians as such or to general students of the theory of statistics in relation to probability. This does not apply, however, to the first section of the first part, entitled Die menschlichen Massenerscheinungen, in which there is a good deal of general theory relating to statistical or inductive probabilities, and in which he supplies a very useful account of the mathematical basis of the statistical methods associated with the name of Prof. Lexis. Probabilities, which are based upon evidence arising out of observed statistical frequencies, Professor Czuber usefully distinguishes under the name of statistical probabilities from those, such as occur in the discussion of games of chance, arising out of a number of alternatives which the principle of non-sufficient reason permits us to regard as equi-probable. Under this heading he deals, by methods very mach like those adopted by numerous other writers, with such questions as the probable limits of unobserved statistical frequencies, given the values of certain similar statistical frequencies which have been observed in the past. I am inclined to think that almost the whole of this is not, as it stands, altogether valid. The probability of an induction cannot be calculated by these precise numerical methods, and some of the conclusions to which they lead seem plainly contrary to what it is reasonable to believe. In order to employ these methods, Professor Czuber has, in fact, to rely upon veiled appeals to the principle of non-sufficient reason at points of the argument, when he is not, apparently, aware that he is doing so, and at points where appeals to this principle cannot be made legitimately. It is the fundamental error of this part of the classical theory of probability that it arrives at far too high a probability in favour of any hypothesis that it is called in to support. J. M. KEYNES.