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On the Further Development of Gompertz's Law

Published online by Cambridge University Press:  18 August 2016

William Matthew Makeham
William Matthew Makeham
Affiliation:
Institute of Actuaries
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Extract

Referring again to Art. 4 of Gompertz's treatise, let us denote by a0dx the actual probability of dying in the infinitely small time dx, at the initial age of the mortality table, and by axdx the abstract chance of death (in the time dx) at the end of x years; that is to say, the chance considered “independently of” (as Gompertz expresses it) or as “abstracted from” the deterioration resulting from increased age, then, according to Gompertz's law, the actual probability of dying, in the time dx, at the latter age will be axqxdx, in which expression q only is an arbitrary constant denoting the rate of deterioration. If, in axqxdx, we put x=0, it becomes a0dx, which coincides with the expression first assumed. Hence, Gompertz's law, which supposes that “the “vital force or recuperative power loses equal proportions in equal “times”, is expressed, in its general form, by the equation μxdx=axqxdx, or, which is the same thing, by μx=axqx.

Type
Research Article
Information
Journal of the Institute of Actuaries , Volume 28 , Issue 4 , April 1890 , pp. 316 - 332
Copyright
Copyright © Institute and Faculty of Actuaries 1890

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References

page 323 note * This reproach will, no doubt; before very long be wiped out, for I observe that among the optional questions at the last Institute Examination (Part II), there are two relating to the theory of errors of observations. (Since this article was in type my attention has been called to Mr. G. F. Hardy's ingenious solution of a “Question in Probabilities”,—J.I.A., xxvii, 214,—which has an important bearing upon the subject here referred to.)

page 327 note * Although the above demonstration is strictly applicable to graduated errors only, it may easily be shown that the result obtained is practically true for integer errors also. For yx being any function of x, we have But in the present case , a function which vanishes both for x=0 and x=∞. Hence, (approximately). (The expression in the text for the mean expected error agrees with that obtained by Mr. Hardy in the note before referred to—J.I.A., xxvii, 214.)