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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2016 • Question 6

Question

Math 6010 • Linear Models Spring 2016 • Question 6

Let \(X_1,X_2,\ldots ,X_n\) be iid exponential(1) random variables and \[ \bar X_n=\frac {1}{n}\sum _{i=1}^n X_i \] be the sample mean. Find two sequences of numbers \(a_n\) and \(b_n\) such that \[ \frac {\log \bar X_n-a_n}{b_n} \] converges in distribution to a standard normal random variable.

Raw LaTeX

Let \(X_1,X_2,\ldots,X_n\) be iid exponential(1) random variables and
\[
  \bar X_n=\frac{1}{n}\sum_{i=1}^n X_i
\]
be the sample mean. Find two sequences of numbers \(a_n\) and \(b_n\) such that
\[
  \frac{\log\bar X_n-a_n}{b_n}
\]
converges in distribution to a standard normal random variable.

Topic Tags

Central Limit Theorems Laws of Large Numbers Multivariate Normal Distributions Weak Convergence and Distributional Limits