Question

Question 3 [25] , Yn denote a random sample of size n from a Let Y, Y2, population with an exponential distribution whose density is given by y > 0 if o, otherwise -E70 cumulative distribution function f(y) L ..,Y} denotes the smallest order statistics, show that Y1) = min{Y1, =nYa) 3.1 show that = nY1) is an unbiased estimator for 0. /12/ /13/ 3.2 find the mean square error for MSE(e). 2 f-llays Iat-k)-at 1-P Question 4[25] 4.1 Distinguish between each of the following concepts in Bayesian Inference framework: conjugate priors vs non-informative /6/ priors. 4.2 Suppose that we conduct independent Bernoulli trials and record Y, the number of the trail on which the first success occurs C The Random variable Y in this case then has a geometric distribution with success probability p. A beta distribution is againa conjugate prior for p. If the chosen Beta prior has parameters a and B, show 4.2.1 that the posterior distribution of ply is Beta with /11/ parameters a*= a+1 and B* = B+ y- 1. Find the Bayes estimators forp and p(1-p). 18/ 4.2.2 A TE MSE = E- 1CP1y-

Answer 1

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