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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2025

Paper 1

A particle arriving at time $s\leq t$ survives until $t$ with probability $e^{-\mu(t-s)}$. Independently retaining each Poisson arrival with this time-dependent probability is <Poisson thinning>. Hence the number alive at time $t$ is Poisson with mean
$$
\lambda\int_0^te^{-\mu(t-s)}ds
=\frac\lambda\mu(1-e^{-\mu t}).
$$
Thus
$$
\boxed{N_t^{\rm alive}\sim\operatorname{Poisson}\left(\frac\lambda\mu(1-e^{-\mu t})\right).}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 28L / c / Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 28L / c / Solution

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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 28L / c / Solution

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