Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2011 • Question 2

Question

Math 6010 • Linear Models Spring 2011 • Question 2

(a) Let \(\Phi \) and \(\phi \) be the standard normal distribution and density functions, respectively. Show that \[ \lim _{x\to \infty }\frac {1-\Phi (x)}{(1/x)\phi (x)}=1. \]

(b) Let \(X_1,X_2,\ldots \) be independent identically distributed standard normal random variables. Show that \[ \limsup _{n\to \infty }\frac {1}{\sqrt {\log n}}|X_n|=c \quad \text {almost surely} \] and compute the value of \(c\).

Raw LaTeX

\textbf{(a)} Let \(\Phi\) and \(\phi\) be the standard normal
distribution and density functions, respectively. Show that
\[
  \lim_{x\to\infty}\frac{1-\Phi(x)}{(1/x)\phi(x)}=1.
\]

\textbf{(b)} Let \(X_1,X_2,\ldots\) be independent identically distributed
standard normal random variables. Show that
\[
  \limsup_{n\to\infty}\frac{1}{\sqrt{\log n}}|X_n|=c
  \quad\text{almost surely}
\]
and compute the value of \(c\).

Topic Tags

Central Limit Theorems Convergence in Probability and Almost Sure Convergence Covariance and Independence Multivariate Normal Distributions