By N. Keyfitz, Hal Caswell
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Extra info for Applied Mathematical Demography, Third Edition (Statistics for Biology and Health)
03 , . . 125 . 1 is worth this extended study because of its important applications. In particular, when r(t) is interpreted as −µ(a), µ(a) being mortality at age a, the result carries over to cohorts; a cohort is deﬁned as a number of individuals born at a particular time and followed through life. Such a cohort is a peculiar population, in that it never receives new members after the initial moment, and its initial births die oﬀ during the 100 or so years of the cohort’s duration. 1 tells us that the survivors to age x, l(x), of a cohort numbering l0 at birth, are equal to x l(x) = l0 exp − µ(a) da .
2), a task that appears hopeless when nothing is known but 5 Mx . 2); this is achieved by making assumptions that somehow restrict p(a). 2) is to assume p(a) = l(a), and also to suppose that l(x) is a straight line. The integral under the straight line is x+5 x Since x+5 x l(a) da = 52 (lx + lx+5 ). 4) from which the value of lx+5 /lx may be obtained by dividing numerator and denominator on the right by lx , and then solving a linear equation for lx+5 /lx to ﬁnd lx+5 1 − 55 Mx /2 . 04147. 1). 5) the probability of dying is 5 qx =1− lx+5 55 Mx .
A useful summarizing device is to consider only one sex at a time. A large part of our work will treat only the female side of the population, or only the male side. This is not to imply that the other sex is not necessary, but rather to suppose that it exists in whatever numbers are required to produce the growth in the sex being followed. The present section is intended to show how this device gives clear answers to questions otherwise indeterminate. We will use the number of descendants of the Pilgrim fathers—not in itself an important demographic issue—as an extreme example of the uncertainty introduced when both sexes are considered simultaneously.