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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2008 • Question 1

Question

Math 6010 • Linear Models Fall 2008 • Question 1

Let \(U_1,U_2,\ldots ,U_n\) be independent random variables uniform on \([0,1]\) and let \(U_{1,n}\leq U_{2,n}\leq \cdots \leq U_{n,n}\) be the corresponding order statistics.

a. Compute the asymptotic distribution of \(nU_{2,n}\). (5 points)

b. Show that \(nU_{1,n}\) and \(n(1-U_{n,n})\) are asymptotically independent. (5 points)

Raw LaTeX

Let \(U_1,U_2,\ldots,U_n\) be independent random variables uniform
on \([0,1]\) and let
\(U_{1,n}\leq U_{2,n}\leq\cdots\leq U_{n,n}\) be the corresponding order
statistics.

a. Compute the asymptotic distribution of \(nU_{2,n}\). (5 points)

b. Show that \(nU_{1,n}\) and \(n(1-U_{n,n})\) are asymptotically independent.
(5 points)

Topic Tags

Covariance and Independence Estimator Distributions Multivariate Normal Distributions Uniform Integrability and Truncation Methods