Solution

Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 4 / 26I / b / Solution

Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 1 / 28K / c / Solution

The pure birth process with rates $\lambda_n=\alpha n+\beta$, where $\alpha,\beta>0$, combines linear reproduction with immigration. Starting from zero, its mean satisfies
$$
m'(t)=\alpha m(t)+\beta,
\qquad
m(0)=0,
$$
and hence
$$
\mathbb EN(t)=\frac{\beta}{\alpha}(e^{\alpha t}-1).
$$


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 Incoming links: Linear birth process with immigration