Exercise6.4 The waitingtimeforabusatagivenbusstopatacertaintimeofdayisknowntohave a uniformdistribution U (0,). Fromothersimilarroutesitisknownthat has theParetodistri- bution Pa(7,4), wherethedensityoftheParetodistribution Pa(,) is f,

Exercise6.4 The waitingtimeforabusatagivenbusstopatacertaintimeofdayisknowntohave a uniformdistribution U (0,θ). Fromothersimilarroutesitisknownthat θ has theParetodistri-

bution Pa(7,4), wherethedensityoftheParetodistribution Pa(α,β) is fα,β (θ) ∝ θ−(α+1), θ ≥ β

(see Exercise1.2).Waitingtimesof10,3,2,5and14minuteshavebeenobservedatthegivenbus stop duringthelast5days.

1. ShowthattheParetodistributionprovidesaconjugatepriorforuniformdataandfindtheposte-

rior distributionof θ.

2. Estimate θ w.r.t.thesquarederror(recallthat E (Pa(α,β))= αβ

α−1 , α > 1).

3. Find a95%HPDfor θ.

4. Testthehypothesis H0 : 0 ≤ θ ≤ 15 vs. H1 : θ > 15 bychoosingthemorelikely(aposteriori)

hypothesis.