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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2014 • Question 9

Question

Math 6010 • Linear Models Fall 2014 • Question 9

Suppose \(X\) and \(Y\) are i.i.d. random variables and \(f\) and \(g\) are nondecreasing, bounded and measurable functions on \(\mathbb {R}\).

(a) Prove that \[ \mathbb {E}[(f(X)-f(Y))(g(X)-g(Y))]\geq 0. \]

(b) Conclude that \(f(X)\) and \(g(X)\) are always positively correlated; i.e., \[ \mathbb {E}[f(X)g(X)]\geq \mathbb {E}[f(X)]\cdot \mathbb {E}[g(X)]. \]

Raw LaTeX

Suppose \(X\) and \(Y\) are i.i.d. random variables and \(f\) and \(g\) are
nondecreasing, bounded and measurable functions on \(\R\).

\textbf{(a)} Prove that
\[
  \E[(f(X)-f(Y))(g(X)-g(Y))]\geq0.
\]

\textbf{(b)} Conclude that \(f(X)\) and \(g(X)\) are always positively correlated;
i.e.,
\[
  \E[f(X)g(X)]\geq \E[f(X)]\cdot\E[g(X)].
\]

Topic Tags

Laws of Large Numbers Measure-Theoretic Probability Uniform Integrability and Truncation Methods