Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

1998 • Question 9

Question

Math 6010 • Linear Models 1998 • Question 9

Let \(X_1,\ldots ,X_n\) be a random sample from \(N(\theta ,\theta )\) (i.e., normal with mean and variance both equal to \(\theta \)), where \(\theta >0\).

(a) Give three pivotal quantities (or pivots) involving the entire sample, which respectively have a standard normal distribution, a \(t\)-distribution, and a chi-squared distribution, and indicate the numbers of degrees of freedom. (5 points)

(b) Use the normal pivotal quantity in part (a) to obtain a \(100(1-\alpha )\) percent confidence interval for \(\theta \). (5 points)

Raw LaTeX

Let \(X_1,\ldots,X_n\) be a random sample from \(N(\theta,\theta)\)
(i.e., normal with mean and variance both equal to \(\theta\)), where \(\theta>0\).

(a) Give three pivotal quantities (or pivots) involving the entire sample, which
respectively have a standard normal distribution, a \(t\)-distribution, and a
chi-squared distribution, and indicate the numbers of degrees of freedom.
(5 points)

(b) Use the normal pivotal quantity in part (a) to obtain a
\(100(1-\alpha)\) percent confidence interval for \(\theta\). (5 points)

Topic Tags

Central Limit Theorems Multivariate Normal Distributions Quadratic Forms and Chi-Squared Distributions Simultaneous Confidence Intervals