Past exam of the physics course of the year 4 of the natural sciences course of the University of Cambridge / 2015 / Paper 1 / Section A / A9 / Solution

Assume independent arrivals at constant rate $r=2000{\mathrm{s}}^{-1}$, so the count follows a Poisson distribution. In $1\mathrm{ms}$ its mean is $\mu =2$. ThereforeOver observation time $T$, the rate estimate is $\hat{r}=N/T$ and its standard deviation is $\sqrt{r/T}$. Thus the fractional one-standard-error uncertainty is $1/\sqrt{rT}$. To make this strictly less than ten per cent requiresThe mean current magnitude is $er\simeq 3.20\times 10^{-16}\mathrm{A}$. The accuracy statement needs a confidence convention: the boxed time uses one standard error, not a guaranteed bound on every realization. A Gaussian 95-percent relative interval would instead require approximately $(1.96/0.1{)}^2$ events, or $T\gtrsim 0.192\mathrm{s}$. Dead time and correlated arrivals would need another count model; the mean rate alone does not establish Poisson statistics.