Past exam of the physics A course of the University of Cambridge / 2018 / Paper 1 / Section D / D12 / c / Solution

Assume independent radioactive detections with negligible dead-time loss, so accumulated counts have a Poisson distribution. The reference measurement has total count $K_0=N_0{t}_0=2018500$ over $t_0=1000\mathrm{s}$. Since a Poisson count has variance equal to its mean, the rate uncertainty isIts relative uncertainty is $7.04\times 10^{-4}$, or $0.0704\%$. For each one-second measurement, ${\sigma }_y\simeq \sqrt{y}$, giving the estimated count and rate uncertaintiesin counts, numerically the same in counts per second for these one-second exposures. Their fractional uncertainties range from about $3.7\%$ at the first sheet to $35$–$45\%$ for the smallest counts. The long, high-rate reference is therefore much more precise. These are statistical errors; a longer exposure does not automatically remove calibration errors, source drift or background bias.