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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2017 • Question 5

Question

Math 6010 • Linear Models Fall 2017 • Question 5

Suppose \(X\) is a nonnegative random variable that satisfies the following: There exist two real numbers \(a,b>0\) such that \(a^n\leq \mathbb {E}[X^n]\leq b^n\) for every integer \(n\geq 1\). Prove that \(\mathbb {P}\{a\leq X\leq b\}=1\).

Raw LaTeX

Suppose \(X\) is a nonnegative random variable that satisfies the
following: There exist two real numbers \(a,b>0\) such that
\(a^n\leq \E[X^n]\leq b^n\) for every integer \(n\geq1\). Prove that
\(\Prob\{a\leq X\leq b\}=1\).

Topic Tags

Measure-Theoretic Probability Uniform Integrability and Truncation Methods