Exercise7.1 The annualnumberofearthquakesinacertainseismicallyactiveregionisdistributed Poisson withanunknownmean . Let Y1, ...,Yn be thenumbersofearthquakesintheareaduring the last n years (itispossibletoassumethatthe Yi

Exercise7.1 The annualnumberofearthquakesinacertainseismicallyactiveregionisdistributed Poisson withanunknownmean λ. Let Y1, ...,Yn be thenumbersofearthquakesintheareaduring the last n years (itispossibletoassumethatthe Yi are independent).Seismologistsbelievethat

λ ∼ Gamma(α,β) and wanttoestimate λ w.r.t.theweightedquadraticloss L(λ,a) = (λ−a)2

λ .

1. Find thecorrespondingBayesestimatorof λ. Whatisitsrisk?Isitadmissible?

2. Find theBayesestimatorfortheprobability p that thefollowingyearwillbe“quiet”(therewill be noearthquakesintheregion)w.r.t.theweightedquadraticloss L(p,a) = (p−a)2 p2 .

3. Are theaboveBayesestimatorsfor λ and p consistent?