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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2015 • Question 7

Question

Math 6010 • Linear Models Spring 2015 • Question 7

Let \(X_1,X_2,\ldots ,X_n\) be a random sample from a \(N(\theta ,\sigma ^2)\) distribution (\(\sigma ^2\) known), and let the prior distribution of \(\theta \) be \(N(\mu _0,\sigma _0^2)\), where \(\mu _0\) and \(\sigma _0^2\) are known.

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

(b) Find the Bayes estimator under squared error loss.

Raw LaTeX

Let \(X_1,X_2,\ldots,X_n\) be a random sample from a
\(\N(\theta,\sigma^2)\) distribution (\(\sigma^2\) known), and let the prior
distribution of \(\theta\) be \(\N(\mu_0,\sigma_0^2)\), where \(\mu_0\) and
\(\sigma_0^2\) are known.

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

\textbf{(b)} Find the Bayes estimator under squared error loss.

Topic Tags

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