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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2011 • Question 9

Question

Math 6010 • Linear Models Spring 2011 • Question 9

Let \(X\geq 0\), \(\mathbb {E} X^2<\infty \), and \(0\leq a<\mathbb {E} X\). Apply the Cauchy–Schwarz inequality to prove that \[ \mathbb {P}(X>a)\geq \frac {(\mathbb {E} X-a)^2}{\mathbb {E} X^2}. \]

Raw LaTeX

Let \(X\geq0\), \(\E X^2<\infty\), and \(0\leq a<\E X\). Apply the
Cauchy--Schwarz inequality to prove that
\[
  \Prob(X>a)\geq\frac{(\E X-a)^2}{\E X^2}.
\]

Topic Tags

Probability Inequalities and Tail Bounds Random Walks and Maximal Inequalities Uniform Integrability and Truncation Methods