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Past exam of the mathematics course of the University of Cambridge

2024

Paper 2

Let $B_n$ count the $+1$ steps. Then
$$
B_n\sim\operatorname{Binomial}(n,p),
\qquad S_n=B_n-(n-B_n)=2B_n-n.
$$
Consequently
$$
\boxed{
\mathbb P(S_n=2k-n)
=\binom nkp^kq^{\,n-k},
\quad k=0,\ldots,n,
}
$$
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>.

Solution

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

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

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

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