Let y 1 , y 2 , y 3 , . . . , y n ,
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Let y1, y2, y3, . . . , yn, represent a random sample of size n selected from a normal probability distribution with unknown mean μ and variance σ2 = 1. Then the sample mean, y̅ , has a normal distribution with mean μ and variance σ2 = 1/n. Assume that the prior distribution for μ is a normal distribution with a mean of 5 and a variance of 1.
a. Find the posterior distribution, g(μ Ι y̅).
b. Using a squared error loss function, show that the Bayesian estimate of μ is a weighted average of y̅ and the mean of the prior distribution, 5.
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Statistics For Engineering And The Sciences
ISBN: 9781498728850
6th Edition
Authors: William M. Mendenhall, Terry L. Sincich
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