Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2020 • Question 10

Question

Math 6010 • Linear Models Spring 2020 • Question 10

Consider the symmetric random variable \(X\) taking values in \(\mathbb {Z}\setminus \{-1,0,1\}\) with distribution \[ \mathbb {P}(X=k)=\mathbb {P}(X=-k)=\frac {C}{k^2\log k}, \qquad k=2,3,\ldots \]

a) Prove that there is a finite \(C<\infty \) such that the above is a probability distribution.

b) Show that \(\mathbb {E}[|X|]=\infty \), hence \(\mathbb {E}[X]\) is not defined.

c) However, prove that \(\phi '_X(0)=0\), which shows that the converse direction of the previous problem is not always true. This requires showing that the limit defining \(\phi '_X(0)\) exists and equals zero. (Hint: Prove then use the inequality \(|\cos x-1|\leq \min (x^2/2,1)\).)

Consider the symmetric random variable \(X\) taking values in
\(\mathbb{Z}\setminus\{-1,0,1\}\) with distribution
\[
  \Prob(X=k)=\Prob(X=-k)=\frac{C}{k^2\log k},
  \qquad k=2,3,\ldots
\]

a) Prove that there is a finite \(C<\infty\) such that the above is a probability
distribution.

b) Show that \(\E[|X|]=\infty\), hence \(\E[X]\) is not defined.

c) However, prove that \(\phi'_X(0)=0\), which shows that the converse direction
of the previous problem is not always true. This requires showing that the
limit defining \(\phi'_X(0)\) exists and equals zero. (\textbf{Hint:} Prove then
use the inequality \(|\cos x-1|\leq\min(x^2/2,1)\).)

Topic Tags

Measure-Theoretic Probability Probability Inequalities and Tail Bounds Random Walks and Maximal Inequalities Uniform Integrability and Truncation Methods