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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.

1

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