By N. Keyfitz, Hal Caswell
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Extra info for Applied Mathematical Demography, Third Edition (Statistics for Biology and Health)
The prospect ahead of any living person is the same, no matter how old he is, on this peculiar table. All the columns of the life table are in general derivable from any one column, and if any column is age-independent all the others referring to a person’s prospects must be also. Application of these deﬁnitions to other mathematically speciﬁed forms o of µ(x) requires more diﬃcult integration to ﬁnd l(x) and e x . 3. Here we proceed to methods for constructing a life table from empirical data.
This is one way of deriving a version of Fisher’s (1930) Fundamental Theorem of Natural Selection. 16 1. Introduction: Population Without Age The argument can be extended to n subpopulations by induction; if Q1 of the n are increasing at ri , and so on, the proof then applies to integral weights Qi ; from there it can be carried to real weights Qi . To present this in detail would carry us too far from demography, which is concerned only with the result that separate projection of each of the various elements of a heterogeneous population gives a total greater than is obtained by projection of the whole population at its average rate.
63x. 63x triplings constitute the same increase. We found that to go from 2 persons to 4 billion persons requires 31 doublings. 63 = 20 triplings. In general, if x is the log of R to base a, and y is the log of R to base b, so that ax = by , then taking logarithms to an arbitrary base gives x log a = y log b, whatever the base. If the base is b, then, since we know (from the fact that b1 = b) that logb b = 1, we have y = x logb a. 8) This says that to go from loga to logb we need merely multiply by logb a.