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 Incoming links: Binomial thinning

Use the zero-based geometric distribution, with

P (N = k) = p (1 - p)^{k}

for

k \geq 0

and

0 < p < 1

0 \leq r \leq 1

, the retained count is again zero-based geometrically distributed, with the displayed success probability. Indeed, writing

q = 1 - p

G_{Y} (s) = \frac{p}{1 - q ( 1 - r + rs )} = \frac{p ^{'}}{1 - ( 1 - p ^{'} ) s} .

For

r = 0

this includes the degenerate success parameter

p^{'} = 1

and retained count zero.

Given

X = k

, the score

Y

1/2

. Its conditional expected value and variance are

k /2

and

k /4

. Thus

E [Y ∣ X = k] = \frac{k}{2}, E [Y^{2} ∣ X = k] = \frac{k}{4} + \frac{k ^{2}}{4} .

This is binomial thinning of the first-round count; the second formula uses the identity

E [Y^{2} ∣ X] = Var (Y ∣ X) + E [Y ∣ X]^{2}