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 Incoming links: Binomial thinning of a geometric random variable

Past exam of the year 1 of the computer science course of the University of Oxford / 2014 / Discrete mathematics, continuous mathematics and probability / 8 / b / ii / Solution 2026-10-31

The law of total expectation gives

E Y = E [X /2] = q / (2 p)

. The law of total variance separates the two sources of randomness:

Var (Y) = E [Var (Y ∣ X)] + Var (E [Y ∣ X]) = \frac{1}{4} E X + \frac{1}{4} Var (X) .

Substitution gives

E Y = \frac{1 - p}{2 p}, Var (Y) = \frac{q ( p + 1 )}{4 p ^{2}} = \frac{1 - p ^{2}}{4 p ^{2}} .

The binomial thinning of a geometric random variable identity supplies a further check:

Y

itself has a zero-based geometric distribution with success probability

2 p / (1 + p)

.

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