Exercise1.2 Find sufficientstatisticsfortheparametersofthefollowingdistributionsbasedona random sampleofsize n: 1. geometric distribution Geom(p) 2. uniform distribution U (, +1) 3.

Exercise1.2 Find sufficientstatisticsfortheparametersofthefollowingdistributionsbasedona random sampleofsize n:

1. geometric distribution Geom(p)

2. uniform distribution U (θ,θ +1)

3. uniform distribution U (−θ,θ)

4. uniform distribution U (θ1,θ2)

5. shifted exponentialdistribution fθ,τ (y) = θe−θ(y−τ), y ≥ τ, θ > 0, where

(a) τ is known

(b) τ is unknown 6. double exponentialdistribution fθ (y) = θ

2 e−θ|y|, −∞ < y < ∞, θ > 0 7. Gamma distribution Gamma(α,β)

8. Beta distribution Beta(p,q)

9. log-normal distribution fμ,σ (y) = √ 1 2πσy e−(ln y−μ)2 2σ2 10. Weibulldistribution fα,β (y) = βαyα−1 exp{−βyα}, y > 0, α > 0 β > 0, where

(a) α is known

(b) α is unknown 11. Paretodistribution fα,β (y) = αβ



β

y

α+1

, y ≥ β, α,β > 0, where

(a) β is known

(b) β is unknown 12. inverseGaussiandistribution fθ1,θ2 (y) =

q

θ1 2πy3 exp n

−

θ1(y−θ2)2 2θ2 2 y o

, y > 0, θ1 > 0 θ2 > 0, where

(a) θ1 is known

(b) θ2 is known

(c) both parametersareunknown