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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2009 • Question 4

Question

Math 6010 • Linear Models Spring 2009 • Question 4

Let \(X_1,\ldots ,X_n\) be a random sample from a \(N(\theta ,1)\) distribution, and let the prior distribution of \(\theta \) be \(N(\mu _0,1)\), where \(\mu _0\) is known.

(a) Find the posterior distribution of \(\theta \).

(b) Find the Bayes estimator of \(\theta \) under squared error loss, and show that it is a weighted average of the prior mean \(\mu _0\) and the sample mean \(\bar X\).

Raw LaTeX

Let \(X_1,\ldots,X_n\) be a random sample from a \(\N(\theta,1)\)
distribution, and let the prior distribution of \(\theta\) be \(\N(\mu_0,1)\),
where \(\mu_0\) is known.

\textbf{(a)} Find the posterior distribution of \(\theta\).

\textbf{(b)} Find the Bayes estimator of \(\theta\) under squared error loss, and
show that it is a weighted average of the prior mean \(\mu_0\) and the sample mean
\(\bar X\).

Topic Tags

Bayesian Testing and Decision Theory Estimator Distributions Quadratic Forms and Chi-Squared Distributions Uniform Integrability and Truncation Methods