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Past exam of the year 1 of the computer science course of the University of Oxford

2023

Discrete mathematics, probability and continuous mathematics

Differentiate the <probability generating function> inside its radius of convergence:
$$
G_N'(s)=\frac{pq}{(1-qs)^2},\qquad G_N''(s)=\frac{2pq^2}{(1-qs)^3}.
$$
The first <moment> and second factorial <moment> are $\mathbb E[N]=G_N'(1)=q/p$ and $\mathbb E[N(N-1)]=G_N''(1)=2q^2/p^2$. Consequently
$$
\boxed{\mathbb E[N]=\frac{1-p}{p},\qquad
\operatorname{Var}(N)=\frac{2q^2}{p^2}+\frac qp-\frac{q^2}{p^2}=\frac{1-p}{p^2}.}
$$

Solution

Past exam of the year 1 of the computer science course of the University of Oxford / 2023 / Discrete mathematics, probability and continuous mathematics / 8 / a / iii / Solution

Past exam of the year 1 of the computer science course of the University of Oxford / 2023 / Discrete mathematics, probability and continuous mathematics / 8 / a / iii / Solution

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