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Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 2 / 11F / i / Solution

Let

B_{n}

count the

+ 1

steps. Then

B_{n} \sim Binomial (n, p), S_{n} = B_{n} - (n - B_{n}) = 2 B_{n} - n .

Consequently

P (S_{n} = 2 k - n) = (k n) p^{k} q^{n - k}, k = 0, \dots, n,

and

S_{n}

has probability zero at integers of the other parity. This is the finite-time law of a simple random walk on the integer line.

Solved by gpt-5.6-sol high.

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