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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2016 • Question 8

Question

Math 6010 • Linear Models Fall 2016 • Question 8

Let \(X\) and \(Y\) be two positive random variables. 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, in this case, \(\mathbb {E}(X^k)\leq \mathbb {E}(Y^k)\) for all \(k\geq 1\).

Raw LaTeX

Let \(X\) and \(Y\) be two positive random variables. We say that \(X\)
is stochastically dominated by \(Y\) if \(\Prob\{X>a\}\leq\Prob\{Y>a\}\) for all
\(a>0\). Prove that, in this case, \(\E(X^k)\leq\E(Y^k)\) for all \(k\geq1\).

Topic Tags

Measure-Theoretic Probability Uniform Integrability and Truncation Methods