Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2013 • Question 6

Question

Math 6010 • Linear Models Fall 2013 • Question 6

For every random vector \((X,Y)\) in \(\mathbf {R}^2\) we can define a function \(\mathcal {Q}\) of two variables as follows: \[ \mathcal {Q}(a,b):=\Prob \{X>a,\ Y>b\}-\Prob \{X>a\}\Prob \{Y>b\}. \]

(a) Suppose \(f,g:\mathbf {R}\to \mathbf {R}\) are bounded and have integrable, bounded, and continuous derivatives. Then prove that \[ \E [f(X)g(Y)]-\E [f(X)]\E [g(Y)] = \iint _{\mathbf {R}^2}\mathcal {Q}(a,b)f'(a)g'(b)\,da\,db. \]

(b) Suppose \(\mathcal {Q}(a,b)\geq 0\) for all \(a,b\in \mathbf {R}\). Prove that if \(X\) and \(Y\) have two finite moments, then \(\operatorname {Cov}(X,Y)\geq 0\) and there exists a finite constant \(\alpha \), depending only on \(f\) and \(g\), such that \[ |\E [f(X)g(Y)]-\E [f(X)]\E [g(Y)]|\leq \alpha \operatorname {Cov}(X,Y). \]

[In other words, if \(\mathcal {Q}\geq 0\) and \(X\) and \(Y\) have a small covariance, then \(X\) and \(Y\) are almost independent.]

For every random vector \((X,Y)\) in \(\mathbf{R}^2\) we can define a
function \(\mathcal{Q}\) of two variables as follows:
\[
  \mathcal{Q}(a,b):=\Prob\{X>a,\ Y>b\}-\Prob\{X>a\}\Prob\{Y>b\}.
\]

\textbf{(a)} Suppose \(f,g:\mathbf{R}\to\mathbf{R}\) are bounded and have
integrable, bounded, and continuous derivatives. Then prove that
\[
  \E[f(X)g(Y)]-\E[f(X)]\E[g(Y)]
  =
  \iint_{\mathbf{R}^2}\mathcal{Q}(a,b)f'(a)g'(b)\,da\,db.
\]

\textbf{(b)} Suppose \(\mathcal{Q}(a,b)\geq0\) for all \(a,b\in\mathbf{R}\). Prove
that if \(X\) and \(Y\) have two finite moments, then \(\Cov(X,Y)\geq0\) and there
exists a finite constant \(\alpha\), depending only on \(f\) and \(g\), such that
\[
  |\E[f(X)g(Y)]-\E[f(X)]\E[g(Y)]|\leq \alpha\Cov(X,Y).
\]

[In other words, if \(\mathcal{Q}\geq0\) and \(X\) and \(Y\) have a small covariance,
then \(X\) and \(Y\) are almost independent.]

Topic Tags

Covariance and Independence L2 Convergence and Covariance Bounds Uniform Integrability and Truncation Methods