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Past exam of the mathematics course of the University of Cambridge

2021

Paper 1

The <moment-generating function> of $W_i\sim\operatorname{Exp}(1)$ is
$$
M_{W_i}(t)=\frac1{1-t},
\qquad t<1.
$$
Independence makes the moment-generating function of the sum equal the product:
$$
M_{\sum_iW_i}(t)=(1-t)^{-n}.
$$
This is the moment-generating function of the <gamma distribution> $\Gamma(n,1)$, and moment-generating functions determine distributions in a neighbourhood of zero. Hence
$$
\boxed{\sum_{i=1}^nW_i\sim\Gamma(n,1)}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 1 / 18H / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 1 / 18H / a / Solution

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Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 1 / 18H / a / Solution

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