Exercise1.1 Let Y1, ...,Yn be arandomsamplefromaPoissondistribution Pois(). InExample1.6, we foundthat T(Y) = ni =1Yi is asufficientstatisticfor

Exercise1.1 Let Y1, ...,Yn be arandomsamplefromaPoissondistribution Pois(λ). InExample1.6, we foundthat T(Y) = Σni

=1Yi is asufficientstatisticfor λ using theFisher–Neymanfactorization theorem.

1. Obtain thisresultdirectlyusingthedefinitionofasufficientstatistic.Whatisthedistributionof T(Y)?

2. ∗Showthat T(Y) is aminimalsufficientstatisticfor λ.

3. ∗Showthat T(Y) is completeusing

(a) the definitionofacompletestatistic.

(b) the propertiesoftheexponentialfamilyofdistributions.