Examples and Problems in Mathematical Statistics

PART III: PROBLEMS

Section 8.1

8.1.1 Let = {B(N, θ);0 < θ < 1} be the family of binomial distributions.

(i) Show that the family of beta prior distributions = {β (p, q); 0 < p, q < ∞} is conjugate to .

(ii) What is the posterior distribution of θ given a sample of n i.i.d. random variables, having a distribution in ?

(iii) What is the predicted distribution of Xn + 1, given (X1, …, Xn)?

8.1.2 Let X1, …, Xn be i.i.d. random variables having a Pareto distribution, with p.d.f.

0 < ν < ∞ (A is a specified positive constant).

(i) Show that the geometric mean, , is a minimal sufficient statistic.

(ii) Suppose that ν has a prior G(λ, p) distribution. What is the posterior distribution of ν given X?

(iii) What are the posterior expectation and posterior variance of ν given X?

8.1.3 Let X be a p–dimensional vector having a multinormal distribution N(μ, ). Suppose that is known and that μ