Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2009 • Question 9

Question

Math 6010 • Linear Models Spring 2009 • Question 9

Suppose \(X_1,X_2,\ldots \) are independent, identically-distributed random variables such that \[ \mathbb {P}\{X_1>x\}= \begin {cases} x^{-\alpha }, & \text {if } x\geq 1,\\ 1, & \text {if } x<1, \end {cases} \] where \(\alpha >0\) is fixed. Prove that \[ n^{-1/\alpha }\max (X_1,\ldots ,X_n)\Rightarrow X, \] and compute \(\mathbb {P}\{X>x\}\) for all \(x\in \mathbf {R}\).

Raw LaTeX

Suppose \(X_1,X_2,\ldots\) are independent, identically-distributed
random variables such that
\[
  \Prob\{X_1>x\}=
  \begin{cases}
    x^{-\alpha}, & \text{if } x\geq1,\\
    1, & \text{if } x<1,
  \end{cases}
\]
where \(\alpha>0\) is fixed. Prove that
\[
  n^{-1/\alpha}\max(X_1,\ldots,X_n)\Rightarrow X,
\]
and compute \(\Prob\{X>x\}\) for all \(x\in\mathbf{R}\).

Topic Tags

Covariance and Independence Measure-Theoretic Probability Uniform Integrability and Truncation Methods