Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2008 • Question 6

Question

Math 6010 • Linear Models Fall 2008 • Question 6

Let \(\Phi \) and \(\phi \) denote the standard normal distribution and density functions.

(a) Prove that for all \(x>0\) \[ 1-\Phi (x)\leq \frac {1}{x}\phi (x). \] (5 points)

Let \(X_{1,n}<X_{2,n}\leq \cdots \leq X_{n,n}\) denote the order statistics from a sample of \(n\) independent identically distributed standard normal random variables.

(b) Show that \(X_{n,n}\) converges in probability to infinity. (5 points)

(c) Show that for all \(\epsilon >0\) \[ \lim _{n\to \infty } \mathbb {P}\left \{X_{n,n}\leq \sqrt {(2+\epsilon )\log n}\right \}=1. \] (5 points)

Let \(\Phi\) and \(\phi\) denote the standard normal distribution and
density functions.

(a) Prove that for all \(x>0\)
\[
  1-\Phi(x)\leq\frac{1}{x}\phi(x).
\]
(5 points)

Let \(X_{1,n}<X_{2,n}\leq\cdots\leq X_{n,n}\) denote the order statistics from a
sample of \(n\) independent identically distributed standard normal random
variables.

(b) Show that \(X_{n,n}\) converges in probability to infinity. (5 points)

(c) Show that for all \(\epsilon>0\)
\[
  \lim_{n\to\infty}
  \Prob\left\{X_{n,n}\leq\sqrt{(2+\epsilon)\log n}\right\}=1.
\]
(5 points)

Topic Tags

Central Limit Theorems Convergence in Probability and Almost Sure Convergence Measure-Theoretic Probability Multivariate Normal Distributions