Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics III / 5 / a

With independent Bernoulli trials of success probability

p

N

counts the failures before the first success. There are

n

failures followed by one success with probability

(1 - p)^{n} p

{0, 1, 2, \dots}

. It counts failures, rather than the total number of trials including the success.

ii

q = 1 - p

G_{N} (s) = E [s^{N}] = n = 0 \sum \infty p (q s)^{n} = \frac{p}{1 - q s}, ∣ s ∣ < q^{- 1} .

The displayed restriction ensures absolute convergence.

G_{N}^{'} (1) = \frac{q}{p}, G_{N}^{''} (1) = \frac{2 q ^{2}}{p ^{2}} .

Since these are

E [N]

and

E [N (N - 1)]

, respectively,

E [N] = \frac{1 - p}{p}, Var (N) = G_{N}^{''} (1) + G_{N}^{'} (1) - G_{N}^{'} (1)^{2} = \frac{1 - p}{p ^{2}} .

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