Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2020 • Question 6

Question

Math 6010 • Linear Models Spring 2020 • Question 6

Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed random variables that have the uniform distribution on \([0,1]\). Let \(X_{1,n}\leq X_{2,n}\leq \cdots \leq X_{n,n}\) be the order statistics. Let \(k\geq 1\) be a fixed integer. Show that \[ Y_n=nX_{k,n} \] converges in distribution and compute the limiting distribution.

Raw LaTeX

Let \(X_1,X_2,\ldots,X_n\) be independent identically distributed
random variables that have the uniform distribution on \([0,1]\). Let
\(X_{1,n}\leq X_{2,n}\leq \cdots\leq X_{n,n}\) be the order statistics. Let
\(k\geq 1\) be a fixed integer. Show that
\[
  Y_n=nX_{k,n}
\]
converges in distribution and compute the limiting distribution.

Topic Tags

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