Let y 1 , y 2 , y 3 , , y n , represent a random sample of size n selected from a normal probability distribution with unknown mean and variance 2 1 Then the sample mean, y , has a normal distribution with mean and variance 2 1 n Assume that the prior distribution for is a normal distribution with a mean of 5 and a variance of 1 a Find the posterior distribution, g( y) b Using a squared error loss function, show that the Bayesian estimate of is a weighted average of y and the mean of the prior distribution, 5

Question: Let y 1 , y 2 , y 3 , . . . , y n , represent a random sample of size n selected

Let y₁, y₂, y₃, . . . , y_n, 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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