Question: 11. When a Poisson process with rate is observed for a time interval of length 'T, the number X of events occurring has the Poisson

11. When a Poisson process with rate is observed for a time interval of length 'T, the number X of events occurring has the Poisson distribution P(~'T) 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 2~T has a X2-distribution with 2r degrees of freedom. For testing H : ;5; ~o at level a one can, under either design, obtain a specified power p against an alternative ~l by choosing 'T and r sufficientlylarge. (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 'T/ r . (ii) Determine for which values of each of the two designs is preferable when ~o = I, ~I = 2, a = .05, P= .9

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