When a Poisson process with rate is observed for a time interval of length , the number X of events occurring has the

When a Poisson process with rate λ is observed for a time interval of length τ , the number X of events occurring has the Poisson distribution P(λτ ). Under an alternative scheme, the process is observed until r events have occurred, and the time T of observation is then a random variable such that 2λT has a χ2-distribution with 2r degrees of freedom. For testing H : λ ≤ λ0 at level

α one can, under either design, obtain a specified power β against an alternative

λ1 by choosing τ and r sufficiently large.

(i) The ratio of the time of observation required for this purpose under the first design to the expected time required under the second is λτ /r.
(ii) Determine for which values of λ each of the two designs is preferable when λ0 = 1, λ1 = 2, α = .05, β = 9.