A Geiger counter is used to measure radioactive decay. To estimate the activity in a radioactive sample one can do the following experiment. First, the natural background radiation is measured during...

A Geiger counter is used to measure radioactive decay. To estimate the activity in a radioactive

sample one can do the following experiment. First, the natural background radiation is measured during a time period Tb and then, during a time period T_s, one measures the sample plus background. Assuming both background and sample radiation are Poisson processes, with intensities λ_b
and λ_s, respectively, explain how the sample activity can be estimated. Also, for a fixed total experiment time
  find the best choice of T_b
and T_s
that makes the variance of the estimate as small as possible. Finally, construct an approximate confidence interval for λ_s.