Let X1, . . . , Xnbe a r.s. from Pareto distribution with density f (x) = βαβ , x > α, α > 0, β > 2.

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

Q 1.  Let X1, . . . , Xnbe a r.s.  from Pareto distribution with density f (x) =  βαβ  , x > α, α > 0, β > 2. Find a sufficient statistic when (i) α is known; (ii) β is known; and

(iii) both α and β are unknown.

Q 2. Let X1, . . . , Xn be a r.s. from Beta(λ, µ) population. Find a sufficient statistic when

(i) µ is known; (ii) λ is known; and (iii) both µ and λ are unknown.

Q 3. Let X1, . . . , Xn be a r.s.  from a continuous population with density function f (x) =   θ , x > 0,  θ > 0. Find a minimal sufficient statistic.

Q 4.  Let  X1, . . . , Xn be  a  r.s.  from  a  Geometric  population  with  pmf  f (x)  =  p(1 − p)x−1, x = 1, 2, . . . ,  0 < p < 1.  Find a minimal sufficient statistic.

Q 5.  Let  X1, . . . , Xn be  a  r.s. from an exponential population with density function

f (x) = eµ−x, x > µ, µ ∈ R. Show that Y = X(1) is complete.

Q 6. Let X1, . . . , Xn be a r.s. from a N (θ, θ2) population. Show that (X, S2) is minimal sufficient but not complete.

Q 7. Let X1, . . . , Xn be a r.s. from a N (0, σ2) population. Show that X is not complete but X2 is complete.

Q 8. Let X1, . . . , Xn be a r.s. from a P (λ) population. Find a UMVUE of g(λ) = P (X1 ≤ 1) = (1 + λ)e−λ.

Q 9.  Let  X1, . . . , Xn be  a  r.s. from an exponential population with density function

f (x) = eµ−x, x > µ,  µ ∈ R.  Find UMVUEs of µ and µ2.

Q 10. Let X1, . . . , Xn be a r.s. from a N (µ, σ2) population. Find UMVUEs of the signal to noise ratio µ/σ and quantile µ + bσ, where b is any real number.

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