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Past exam of the mathematics course of the University of Oxford

2017

Prelims

Mathematics III

{c}

Using the <geometric series>, the <probability generating function> is
$$
G_X(s)=\mathbb E(s^X)=\sum_{k\ge1}pq^{k-1}s^k
=\boxed{\frac{ps}{1-qs}},\qquad |qs|<1.
$$
Differentiation gives
$$
\boxed{G'_X(s)=\frac{p(1-qs)+pqs}{(1-qs)^2}=\frac{p}{(1-s+ps)^2}.}
$$

Solution

Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics III / 4 / a / ii / Solution

Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics III / 4 / a / ii / Solution

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Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics III / 4 / a / ii / Solution

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