Solution
= Solution
The number probabilities are
$$
\mathbb P(N=n)=e^{-|z|^2}\frac{|z|^{2n}}{n!},
$$
a <Poisson distribution> of parameter $|z|^2$. Hence
$$
\boxed{\langle N\rangle=|z|^2,\qquad
(\Delta N)^2=|z|^2,\qquad \Delta N=|z|}.
$$
The relative uncertainty is $1/|z|=1/\sqrt{\langle N\rangle}$, which tends to zero as the mean occupation tends to infinity.