= Solution
For the Gaussian approximation,
$$
\langle n^2\rangle\simeq\mu^2+\sigma^2
=\frac{\lambda}{2\beta}+\frac{3\mu}{4}.
$$
The extra term reflects the subleading error made by centring the linear approximation at the deterministic fixed point. It is smaller than $\mu^2$ by relative order $1/\mu$, so the exact steady-state condition from part (a) is recovered at leading order. A higher-order expansion shifts the mean and restores the moment balance more accurately.
The approximation is reliable when
$$
\mu=\sqrt{\frac{\lambda}{2\beta}}\gg1,
\qquad
\frac{\sigma}{\mu}=O(\mu^{-1/2})\ll1.
$$
\b[Thus one needs $\lambda/\beta\gg1$: births must support a large typical population, with jumps of one or two individuals small compared with the population scale.]
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