TFR poisson (PDF)

We want to calculate the expected value of siblings, if the number of children is poisson
distributed with parameter λ. Let X be the random variable associated with the number of
siblings. Assume there are n women. If n is very large, the sample mean
P∞ converges to
m ∗k∗(k−1)
P∞ k
the true mean of the random variable. The sample mean is given by k=0
.
k=0 mk ∗k
Here mk denotes the number of women brithing k siblings. The formula holds because the
number of individuals in the sibling sample is simply givent by the denominator (mothers
times number of ofspring) and the (k-1) in the numerator comes from each of the k children
having k − 1 siblings.
Now heuristically we can assume for large n that mk is approximately n ∗ pk , since
this means that the probability for a single mother to have k kids is pk in the population. The n cancels in the fraction and the denominator is simply the mean of the poisson
numerator is aPlittle more tricky: Splitting the sum gives us
P∞distribution, i.e λ. The
P∞
∞
2
m
∗
k
∗
(k
−
1)
=
k
k=0
k=0 mk ∗ k −
k=0 mk ∗ k. The second summand is λ again.
The first one is the second Moment of the Poisson distribution, i.e λ2 + λ. Subtracting λ
from it yields the result. The more interresting takeaway is that for any discrete distribution
with suffisciently nice properties the above heuristic probably works and you get that the
expected number of siblings is E[Y 2 ]/E[Y ] − 1, where Y is the random variable ofspring
is drawn from.

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