method to frequency-loci of two dimensions;[1] constructing for the curve of regression (as a substitute for the normal right line), in the case of “skew correlation,” a parabola,[2] with constants based on the higher moments of the given group.
168. In this connexion reference may again be made to Mr Yule’s method of treating skew surfaces as if they were normal. It is certainly remarkable that the correlation should be so well represented by a line—the property of a normal surface—in cases of which normality cannot be predicated: for instance, the statistics of the number of husbands (or wives) living at each age who have wives (or husbands) living at different ages.[3] It may be suggested that though in this case there is one dominant cause, the continual decrease of the population, inconsistent with the plurality of causes postulated for the law of error, yet there is a sufficient degree of accidental variation to realize one property at least of the normal locus.
169. There is possibly an extensive class of phenomena of which frequency depends largely on fortuitous causes, yet not so completely as to present the genuine law of error.[4] This mixed class of phenomena might be amenable to a kind of law of frequency that would be different Relations between Frequency and Probability. from, yet have some affinity to, the law of error. The double character may be taken as the definition of the laws proper to the present section. The definition of the class is more distinct than its extent. Consider for example the statistics which represent the numbers out of a million born that die in each year of age after thirty of forty—the latter part of the column in a life-table. These are well represented by a species of Professor Pearson’s “generalized probability-curve,”[5] his type iii. of the form
y = y0(1 + x/a)γαe−−γχ.
The statistics also lend themselves to the Gompertz-Makeham formula for the number living at the age
lx = Sxgx/c.
The former law, the simplest species of the “generalized probability-curve,” may well be attributed in part to the operation of a plexus of causes such as that which is apt to generate the law of error. In fact, a high authority, Professor Lexis, has seen in these statistics—or continental statistics in pari materia—a fulfilment of the normal law of error.[6] They at least fulfil tolerably the generalized law of error above described. But the Gompertz-Makeham formula is not thus to be accounted for; at least it is not thus that it was regarded by its discoverers. Gompertz justifies his law[7] by a “hypothetical deduction congruous with many natural effects,” such as the exhaustion of air by a pump; and Makeham follows[8] in the same track of explanation by way of natural laws. Of course it is not denied that mortality is subject to accident. But the Gompertz-Makeham law purports to be fulfilled in spite of, not by reason of, fortuitous agencies. The formula is accounted for not by the interaction of fleeting causes which is characteristic of probability, but by causes of that ordinary kind of which the investigation constitutes the greater part of natural science. Laws of frequency thus conceived do not belong to the theory of Probabilities.
Authorities.—As a comprehensive and masterly treatment of the subject as a whole, in its philosophical as well as mathematical character, there is nothing similar or second to Laplace’s Théorie analytique des probabilités. But this “ne plus ultra of mathematical skill and power” as it is called by Herschel (Edinburgh Review, 1850) is not easy reading. Much of its difficulty is connected with the use of a mathematical method which is now almost superseded, “Generating Functions.” Not all parts of the book are as rewarding as the Introduction (published separately as Essai philosophique des probabilités) and the fourth and subsequent chapters of the second book. Among numerous general treatises E. Czuber’s Wahrscheinlichkeitstheorie (1899) may be noticed as terse, lucid and abounding in references. Other authorities may be mentioned in relation to the different parts of the subject as above divided. First principles are discussed with remarkable acumen by J. Venn in Logic of Chance (1st ed., 1876, 3rd ed., 1888) and by J. v. Kries in Principien der Wahrscheinlichkeitsrechnung (1886). As a repertory of neat problems involving the calculation of probability and expectation W. A. Whitworth’s Choice and Chance (5th ed., 1901), and DCC. Exercises . . . in Choice and Chance (1897) deserve mention. But this advantage is afforded in nearly as great perfection by more comprehensive works. Bertrand’s Calcul des probabilités (1889) abounds in choice examples, while it excels in almost every other branch of the subject. Special mention is also deserved by H. Poincaré’s Calcul des probabilités (leçons professes, 1893–1894). On local or geometrical probability Professor Morgan Crofton is one of the highest authorities. His paper on “Local Probability” in Phil. Trans. (1868), and on “Geometrical Theorems,” Proc. Lond. Math. Soc. (1887), viii., should be read in connexion with the section on “Local Probability” in his article on “Probability” in the 9th edition of the Ency. Brit., from which section several paragraphs have been transferred en bloc to the section on Geometrical Applications in the present article. The topic is treated exhaustively by Czuber in Geometrische Wahrscheinlichkeiten und Mittelworten (1884). Czuber is also to be mentioned as the author of Theorie der Beobachtungsfehler, in which he has reproduced, often with improvement, or referred to, almost everything of importance in the work of his predecessors. A. L. Bowley’s Elements of Statistics, pt. 2 (2nd ed., 1902), forms an introduction to the law of error which leads the beginner easily, yet far. References to other writers are given in Section I. of Part II. above. A list of writings on the cognate topic, the method of least squares, has been given by Merriman (Connecticut Trans. vol. iv.). On laws of frequency, as above defined, Professor Karl Pearson is the highest authority. His “Contributions to the Mathematical Theory of Evolution,” of which twelve have appeared in the Trans. Roy. Soc. (1894–1903) and others are being published by the Drapers' Company, teem with new theories in Probabilities. (F. Y. E.[9])
PROBATE, in English law, the “proving” (Lat. probatio) of a will. The early jurisdiction of the English ecclesiastical courts over the probate of wills of personality is discussed under Will. The Court of Probate Act 1857 transferred the jurisdiction both voluntary and contentious of all ecclesiastical, royal peculiar, peculiar and manorial courts to the court of probate thereby constituted, created a judge and registrars of that court, abolished the old exclusive rights in testamentary matters of the advocates of Doctors’ Commons, and laid down rules of procedure. Contentious jurisdiction was given to county courts when the personal estate of the deceased was under £200 in value. The Judicature Act 1873 merged the old court of probate in the probate divorce and admiralty division of the High Court of justice. The division now consists of the president and one other judge. The practice of the division is mainly regulated by the rules of the Supreme Court 1883. Appeals lie to the court of appeal and thence to the House of Lords. Probate may be taken out either in common or solemn form. In the former case, which is adopted when there is no dispute as to the validity of the will, the court simply recognizes the will propounded as the last will of the deceased. This formality is necessary to enable the executor to administer the estate of his testator. Probate in this form is granted simply as a ministerial act if the attestation clause declares that the formalities of the Wills Act have been complied with, or if other evidence to that effect is produced. Such grant is liable to revocation, but it is provided that any person dealing with an executor on the faith of a grant of probate in common form, shall not be prejudiced by its revocation. The executor may within thirty years be called upon to prove in solemn form, or a person who doubts the validity of the will propounded may enter a caveat which prevents the executor proving for six months and the caveat may be renewed each six months. The executor may however take out a summons to get the caveat “subducted” or withdrawn, but if an appearance to the summons is entered
- ↑ “Contributions,” No. xiv. (above cited).
- ↑ Not the same parabola as that proposed at par. 162.
- ↑ Census of England and Wales General Report (cod. 2174), p. 226. Cf. p. 70, as to the rationale of the phenomenon.
- ↑ A good example of the suggested blend between law and chance is presented by an hypothesis which Benine (in a passage referred to above, par. 97) has proposed to account for Pareto’s income-curve.
- ↑ “Contributions,” No. ii., Phil. Trans. (1895), vol. 186, A.
- ↑ Lexis, Massenerscheinungen, § 46. Cf. Venn, cited above, par. 124.
- ↑ Phil. Trans. (1-25).
- ↑ Assurance Magazine (1866), xi. 315.
- ↑ These initials do not apply to certain passages in the above article, namely, the greater part of paragraphs 41, 52, 62 and 72, and almost the whole of the 4th section of Part. I. (pars. 76-93), which have been adopted from the article “Probability” in the 9th edition of the Ency. Brit., written by Professor Morgan Crofton.
