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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2019 • Question 3

Question

Math 6010 • Linear Models Spring 2019 • Question 3

Let \(X\) and \(Y\) be two random variables such that \(\mathbb {P}(X\leq Y)=1\). Assume also that \(\mathbb {P}(X\leq x)\leq \mathbb {P}(Y\leq x)\) for all \(x\in \mathbb {R}\). Prove that \(\mathbb {P}(X=Y)=1\).

Raw LaTeX

Let \(X\) and \(Y\) be two random variables such that
\(\Prob(X\leq Y)=1\). Assume also that \(\Prob(X\leq x)\leq\Prob(Y\leq x)\) for
all \(x\in\mathbb{R}\). Prove that \(\Prob(X=Y)=1\).

Topic Tags

Measure-Theoretic Probability Uniform Integrability and Truncation Methods