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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2005 • Question 4

Question

Math 6010 • Linear Models 2005 • Question 4

Let \(X\) be a mean-zero random variable with \(\mathbb {E}(X^2)=\sigma ^2<\infty \). Prove Cantelli’s inequality: For all \(\lambda >0\), \[ \mathbb {P}\{X>\lambda \}\le \inf _{t>0}\left [\frac {t^2+\sigma ^2}{(t+\lambda )^2}\right ] =\frac {\sigma ^2}{\sigma ^2+\lambda ^2}. \] Prove that this is a better bound than Chebyshev’s inequality.

Raw LaTeX

Let \(X\) be a mean-zero random variable with
\(\E(X^2)=\sigma^2<\infty\). Prove Cantelli's inequality: For all \(\lambda>0\),
\[
  \Prob\{X>\lambda\}\le
  \inf_{t>0}\left[\frac{t^2+\sigma^2}{(t+\lambda)^2}\right]
  =\frac{\sigma^2}{\sigma^2+\lambda^2}.
\]
Prove that this is a better bound than Chebyshev's inequality.

Topic Tags

Borel-Cantelli Lemmas Probability Inequalities and Tail Bounds Random Walks and Maximal Inequalities Uniform Integrability and Truncation Methods