Radioactive Decay — Question 5

Question 5

Background subtraction increases the counting uncertainty

A sample-plus-background run records G=2500 counts in 100 s. An independent background-only run records B=400 counts in the same duration. Both counts are Poisson distributed; source decay is negligible during measurement.

1. Find the net count rate and its one-standard-deviation uncertainty.

2. If both acquisition times are increased by a factor of four at unchanged rates, find the new relative uncertainty.

Question 5 - Solution

Background subtraction increases the counting uncertainty

(a) Method and calculation

$$\boxed{R_s=(G-B)/T=(2500-400)/100=21.0\,\mathrm{s^{-1}}}.$$$$\boxed{\sigma_{R_s}=\sqrt{G+B}/T=\sqrt{2900}/100=.5385\,\mathrm{s^{-1}}}.$$

(b) Interpretation and further calculation

$$\boxed{\sigma_{R_s}/R_s=\sqrt{2900}/2100=.02564}.$$$$T\to4T:\quad\boxed{(\sigma_R/R)_{\mathrm{new}}=.02564/2=.01282=1.282\%}.$$

Physical check

The background mean is subtracted, but its variance is added because it was measured independently. Subtracting uncertainties would underestimate the error.