Nahuatl names and natality in aztec empire

[Davius Sanctex]

A nice mathematical argument relating name "Teyacapan"
to average number for children in Aztec Empire.
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Bob McCaa points that in a census of 1540 in a group
of 1205 women, 313 were named "Teyacapan" (= first
born), i.e. 25,04%:



I will show that these data implies
that the average number per family was at most 5,135
and that 6,88% of couples had not any child!

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This number can be related to the average number of
children in a family. We assume:

1) Nearly all first born female babies was named
   "Teyacapan"
2) The probability of borns by unit of time remain
   uniform for a community and population is stationary.

First step:
Second hypothesis implies borns can be well modelized by
a Poisson distribution, thus the probability of a couple
to have k kids is P(k):

P(k)= exp(-m)*(m^k)/k!

[Where m is the average number of children]

Second step:
Thus if the probability of a child to belong to a familiy
with exactly k kids is p(k):

p(k) = P(k)/(1-P(0))

[p(0) is the % of families that have no kid].

Thirst step:
If we take a woman at random the probability of being the
first kid in a family of k kids is just 1/k [= q(k)].
And thus the probability that a woman to be the first baby
of a family is r:

r = p(1)*q(1) + p(2)*q(2) + ...+ p(k)*q(k) + ... =
  = P(1)/1 + p(2)/2 + ...+ p(k)/k + ... = 25,04%

This last equation enable us to evaluate m. For m = 5,135

k    P(k)     p(k)   p(k)/k

0  0,00588     __
1  0,03022  0,03039  0,03039
2  0,07759  0,07805  0,03902
3  0,13282  0,13361  0,04453
4  0,17053  0,17154  0,04288
5  0,17515  0,17618  0,03523
6  0,14991  0,15079  0,02513
7  0,10997  0,11063  0,01580
8  0,07059  0,07101  0,00887
9  0,04028  0,04052  0,00450

sum                  0,2498 = 24,98%

This shows that the average number must be of order 5,135.
Moreover, of this table we deduce that 5,88 % = P(0) of
couples have not babies and the majority (17,51%) have 5
babies. The number of families with 9 is 4,02% ...
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