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Natural sciences course of the University of Cambridge

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Year 1 of the natural sciences course of the University of Cambridge

Mathematics course of the year 1 of the natural sciences course of the University of Cambridge

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge

2017

Paper 2

The zero-goal term contributes nothing to the <expected value>. Shift the summation index in the remaining terms:
$$
\begin{aligned}
\mu&=e^{-\lambda}\sum_{k=1}^\infty k\frac{\lambda^k}{k!}
=\lambda e^{-\lambda}\sum_{k=1}^\infty\frac{\lambda^{k-1}}{(k-1)!}\\
&=\lambda e^{-\lambda}e^\lambda=\boxed{\lambda}.
\end{aligned}
$$
The exponential series justifies the convergent reindexing. Thus the parameter is the <mean> goals per game.

Solution

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2017 / Paper 2 / 14 / b / iii / Solution

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2017 / Paper 2 / 14 / b / iii / Solution

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2017 / Paper 2 / 14 / b / iii / Solution

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