A physicist wants to estimate the rate of emissions of alpha particles from a certain source. He makes two counts. First, he measures the background rate by counting the number of particles in 100 seconds in the absence of the source. He counts 36 background emissions. Then, with the source present, he counts 324 emissions in 100 seconds. This represents the sum of source emissions plus background emissions.

Question

a. Estimate the background rate, in emissions per second, and find the uncertainty in the estimate.
b. Estimate the sum of the source plus background rate, in emissions per second, and find the uncertainty in the estimate.
c. Estimate the rate of source emissions in particles per second, and find the uncertainty in the estimate.
d. Which will provide the smaller uncertainty in estimating the rate of emissions from the source: (1) counting the background only for 150 seconds and the background plus the source for 150 seconds, or (2) counting the background for 100 seconds and the source plus the background for 200 seconds? Compute the uncertainty in each case.
e. Is it possible to reduce the uncertainty to 0.03 particles per second if the background rate is measured for only 100 seconds? If so, for how long must the source plus background be measured? If not, explain why not.

Answer 1

Let represent the background rate, let represent the source rate, and let represent the total rate, all in particles per second.



(a) Let X be the number of background events counted in 100 seconds. Then X - Poisson(100). The observed value of is 36. The estimated background rate is

= x/100= 36/100 = 0.36. The uncertainty is Replacing



(b) Let be the number of events, both background and source, counted in 100 seconds. Then.

The observed value of Y is 324.

The estimated total rate is The uncertainty is Replacing yields





(c)



Now are independent, because they are based on independent measurements ( X and Y )





(d) If background and source plus background are each counted for 150 seconds, then



If background is counted for 100 seconds and source plus background is counted for 200 seconds, then



Counting background for 100 seconds and source plus back-ground for 200 seconds produces the smaller uncertainty



(e) It is not possible. Assume that the background emissions are counted for 100 seconds, and the source plus background is counted for N seconds. Then the uncertainty in the estimate of the source rate is No matter how large N is, this quantity is always greater than and thus greater than 0.003.