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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Summer 2020 • Question 1

Question

Math 6010 • Linear Models Summer 2020 • Question 1

Let \(U_1,U_2,\ldots ,U_n\) be independent and identically distributed random variables, uniform on \([0,1]\). Let \(U_{1,n}\leq U_{2,n}\leq \cdots \leq U_{n,n}\) be the order statistics. Show that \(nU_{3,n}\) converges in distribution and compute the limit.

Raw LaTeX

Let \(U_1,U_2,\ldots,U_n\) be independent and identically distributed
random variables, uniform on \([0,1]\). Let
\(U_{1,n}\leq U_{2,n}\leq \cdots\leq U_{n,n}\) be the order statistics. Show that
\(nU_{3,n}\) converges in distribution and compute the limit.

Topic Tags

Covariance and Independence Estimator Distributions Uniform Integrability and Truncation Methods Weak Convergence and Distributional Limits