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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2009 • Question 2

Question

Math 6010 • Linear Models Fall 2009 • Question 2

Let \(X\) be a random variable with values in an interval \(I\), and suppose that \(f\) and \(g\) are nondecreasing functions on \(I\) such that \(f(X)\) and \(g(X)\) have finite variance. Show that \(\operatorname {Cov}(f(X),g(X))\geq 0\).

Hint: If \(X_1\) and \(X_2\) are i.i.d. as \(X\), then \((f(X_1)-f(X_2))(g(X_1)-g(X_2))\geq 0\).

Raw LaTeX

Let \(X\) be a random variable with values in an interval \(I\), and
suppose that \(f\) and \(g\) are nondecreasing functions on \(I\) such that \(f(X)\)
and \(g(X)\) have finite variance. Show that \(\Cov(f(X),g(X))\geq0\).

Hint: If \(X_1\) and \(X_2\) are i.i.d. as \(X\), then
\((f(X_1)-f(X_2))(g(X_1)-g(X_2))\geq0\).

Topic Tags

Analysis of Variance Measure-Theoretic Probability Simultaneous Confidence Intervals Uniform Integrability and Truncation Methods