Cantelli's inequality

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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:

Step 1

One sets


Then first is

and further


step 2

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 :

step 3

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

  1. Klaus D. Schmidt: Measure and probability. 2009, pp. 288-289
  2. For a real random variable , its expected value is designated.
  3. For a real random variable , its variance is designated.