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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2010 • Question 3

Question

Math 6010 • Linear Models Spring 2010 • Question 3

Suppose \(X_1,X_2,\ldots \) are independent, identically-distributed exponential random variables with mean \(\lambda >0\). Prove that \[ \max (X_1\ldots ,X_n)-\frac {1}{\lambda }\ln n\Rightarrow X, \] and compute \(\mathbb {P}\{X>x\}\) for all \(x>0\).

Raw LaTeX

Suppose \(X_1,X_2,\ldots\) are independent, identically-distributed
exponential random variables with mean \(\lambda>0\). Prove that
\[
  \max(X_1\ldots,X_n)-\frac{1}{\lambda}\ln n\Rightarrow X,
\]
and compute \(\Prob\{X>x\}\) for all \(x>0\).

Topic Tags

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