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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/ia/paper-2/3f/b/solution

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Past exam of the mathematics course of the University of Cambridge / 2003 / ia / Paper 2 / 3F / b / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-14

For independent random variables, their sum has the product of their probability generating functions:

G_{X + Y} (z) = E [z^{X} z^{Y}] = G_{X} (z) G_{Y} (z) = (1 - p + p z)^{n + m} .

(1)

Comparing polynomial coefficients identifies a binomial distribution with parameters

n + m, p

P (X + Y = k) = (k n + m) p^{k} (1 - p)^{n + m - k}, k = 0, \dots, n + m,

(2)

and the probability is zero at all other values. The common success probability is essential to this binomial form; independent binomial counts with different probabilities do not in general add to a binomial count.

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