 Exposure time for a Poisson rate at fixed fractional precision

For an independent count with a Poisson distribution of mean $RT$, the rate estimator $N/T$ has variance $R/T$. Its fractional standard deviation is $1/\sqrt{RT}$, so achieving target fractional precision $ϵ$ requires expected count $RT={ϵ}^{-2}$ and exposure $T=1/({ϵ}^{2}R)$.

For independently acquired settings with rates $R_i$, the required total exposure is ${ϵ}^{-2}{\sum }_i{R}_i^{-1}$. Rare settings dominate because they require the same expected count but collect it more slowly. Simultaneous acquisition is a different design, and backgrounds or correlated systematic errors require a different uncertainty calculation.