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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2010 • Question 1

Question

Math 6010 • Linear Models Spring 2010 • Question 1

In this problem, all random variables are nonnegative. We say that \(X\) is stochastically dominated by \(Y\) if \(\mathbb {P}\{X>a\}\leq \mathbb {P}\{Y>a\}\) for all \(a>0\). Prove that if \(X\) is stochastically dominated by \(Y\), then \(\mathbb {E}\Phi (X)\leq \mathbb {E}\Phi (Y)\) for all increasing functions \(\Phi :\mathbf {R}_+\to \mathbf {R}_+\).

Raw LaTeX

In this problem, all random variables are nonnegative. We say that
\(X\) is stochastically dominated by \(Y\) if \(\Prob\{X>a\}\leq\Prob\{Y>a\}\) for
all \(a>0\). Prove that if \(X\) is stochastically dominated by \(Y\), then
\(\E\Phi(X)\leq\E\Phi(Y)\) for all increasing functions
\(\Phi:\mathbf{R}_+\to\mathbf{R}_+\).

Topic Tags

Measure-Theoretic Probability Uniform Integrability and Truncation Methods