1. The proportion prior probability 1 B(a, b) With corresponding mean and variance, () of...
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1. The proportion prior probability 1 B(a, b) With corresponding mean and variance, α π(θ) of defective items in a large shipment is unknown, but a beta density function of the form = -0⁰-1 (1-0)B-1, 0 <0 <1 and B(a, b) = E (0) = = a + ß and var (0) = αβ (a + b)²(a + B + 1) Γ(α)Γ(β) I(a + B) a) Given that the prior mean E (0) and standard deviation (0) are both 10-², (i) use the formula of E (0) to find an expression for p in terms of a. (ii) Substitute this into the formula of var (0) to evaluate the hyperparameters a and p. b) Given that 100 items are selected at random from the shipment and 3 of these are found to be defective. Determine the posterior probability density function of 0 [Hints: observations have the Binomial distribution] c) Find the Bayesian estimate of 0 under the quadratic loss function. 1. The proportion prior probability 1 B(a, b) With corresponding mean and variance, α π(θ) of defective items in a large shipment is unknown, but a beta density function of the form = -0⁰-1 (1-0)B-1, 0 <0 <1 and B(a, b) = E (0) = = a + ß and var (0) = αβ (a + b)²(a + B + 1) Γ(α)Γ(β) I(a + B) a) Given that the prior mean E (0) and standard deviation (0) are both 10-², (i) use the formula of E (0) to find an expression for p in terms of a. (ii) Substitute this into the formula of var (0) to evaluate the hyperparameters a and p. b) Given that 100 items are selected at random from the shipment and 3 of these are found to be defective. Determine the posterior probability density function of 0 [Hints: observations have the Binomial distribution] c) Find the Bayesian estimate of 0 under the quadratic loss function.
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a i E0 001 B99a ii 001 a 99a 100a2 100a1 a B 098 B 9702 b poste... View the full answer
Related Book For
Applied Regression Analysis and Other Multivariable Methods
ISBN: 978-1285051086
5th edition
Authors: David G. Kleinbaum, Lawrence L. Kupper, Azhar Nizam, Eli S. Rosenberg
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