# Tutprac7 - θ(b If the loss function is w y-θ 2 ﬁnd the Bayesian point estimate w y of θ(c Show that this w y is a weighted average of the

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620-202 Tutorial 7. 1. In a clinical trial, let the probability of a successful outcome have a prior distri- bution that is uniform over [0 , 1]. Suppose that the ﬁrst patient has a successful outcome. Find the Bayes estimate of θ that would be obtained for the squared error loss. 2. (7.2-1) Let Y be the sum of the observations from a Poisson distribution with mean θ . Let the prior p.d.f. of θ be gamma with parameters α and β so that f ( θ ) = 1 Γ( α ) β α θ α - 1 e - θ/β , 0 θ < (a) Find the posterior p.d.f. of θ given Y = y . (Hint: You should be able to recognise the form of the numerator so only consider the terms that involve
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Unformatted text preview: θ ). (b) If the loss function is [ w ( y )-θ ] 2 ﬁnd the Bayesian point estimate w ( y ) of θ . (c) Show that this w ( y ) is a weighted average of the maximum likelihood esti-mate y/n and the prior mean αβ , with respective weights n/ ( n + 1 /β ) and (1 /β )( n + 1 /β ). 3. (7.2-4) Consider a random sample X 1 ,...,X n from a distribution with p.d.f. f ( x | θ ) = 3 θx 2 e-θx 3 , < x < ∞ Let θ have a prior p.d.f. which is gamma with α = 4 and β = 1 / 4. Find the conditional mean of θ , given X 1 = x 1 ,...,X n = x n . 1...
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## This note was uploaded on 10/05/2009 for the course STATS 620-202 taught by Professor R during the One '09 term at University of Melbourne.

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