The inequality of Cantelli is an elementary stochastic inequality that the Italian mathematician Francesco Paolo Cantelli back. It is related to the Chebyshev-Markov inequality and provides a one-sided estimate of the probability that a real random variable will exceed its expected value by a positive number .
Formulation of the inequality
The Cantellian inequality can be stated as follows:
- A probability space and a real random variable are given .
have a finite second moment :
- Let a real number be given and denote the variance of .
Then there is the inequality
Proof of the inequality
Following the presentation by Klaus D. Schmidt , it can be derived as follows:
Then first is
If one now has a (initially arbitrary) real number , the following chain of inequalities results, especially because of the Chebyshev-Markov inequality for second moments :
Especially for the real number
applies after step 2:
So everything is proven.
The real-valued function occurring in step 2 above
takes at the named place
their bare minimum . The upper bound mentioned in the Cantellian inequality is therefore optimal in this sense.
- Klaus D. Schmidt: Measure and probability (= Springer textbook ). Springer Verlag, Berlin, Heidelberg 2009, ISBN 978-3-540-89729-3 .
References and footnotes
↑ Klaus D. Schmidt: Measure and probability. 2009, pp. 288-289
↑ For a real random variable , its expected value is designated.
↑ For a real random variable , its variance is designated.