# Cantelli's inequality

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   .${\ displaystyle (\ Omega, {\ mathcal {A}}, \ operatorname {P})}$ ${\ displaystyle X \ colon (\ Omega, {\ mathcal {A}}, \ operatorname {P}) \ to \ mathbb {R}}$ ${\ displaystyle X}$ have a finite second moment :
${\ displaystyle \ operatorname {E} (X ^ {2}) <{\ infty}}$ .
Let a real number be     given and denote the variance of .${\ displaystyle c> 0}$ ${\ displaystyle \ operatorname {V} (X)}$ ${\ displaystyle X}$ Then there is the inequality
${\ displaystyle \ operatorname {P} {{\ bigl (} X \ geq {\ operatorname {E} {(X)} + c} {\ bigr)}} \ leq {\ frac {\ operatorname {V} {{ \ bigl (} X {\ bigr)}}} {c ^ {2} + {\ operatorname {V} (X)}}}}$ .

## Proof of the inequality

Following the presentation by Klaus D. Schmidt , it can be derived as follows:

### Step 1

One sets

${\ displaystyle Z = X - {\ operatorname {E} (X)}}$ .

Then first is

${\ displaystyle \ operatorname {E} (Z) = 0}$ and further

${\ displaystyle \ operatorname {V} (Z) = \ operatorname {V} (X) = \ operatorname {E} {\ bigl (} Z ^ {2} {\ bigr)}}$ .

### 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 : ${\ displaystyle t> -c}$ {\ displaystyle {\ begin {aligned} \ operatorname {P} {{\ bigl (} X \ geq {\ operatorname {E} {\ bigl (} X {\ bigr)} + ​​c} {\ bigr)}} & = \ operatorname {P} {\ bigl (} Z \ geq c {\ bigr)} \\ & = \ operatorname {P} {\ bigl (} Z + t \ geq c + t {\ bigr)} \\ & \ leq \ operatorname {P} {\ bigl (} | Z + t | \ geq c + t {\ bigr)} \\ & \ leq {\ frac {\ operatorname {E} {\ bigl (} {(Z + t)} ^ {2} {\ bigr)}} {{(c + t)} ^ {2}}} \\ & = {\ frac {\ operatorname {E} {\ bigl (} Z ^ {2} {\ bigr)} + ​​t ^ {2}} {{(c + t)} ^ {2}}} \\ & = {\ frac {{\ operatorname {V} (Z)} + t ^ {2} } {(c + t) ^ {2}}} \\ & = {\ frac {{\ operatorname {V} (X)} + t ^ {2}} {(c + t) ^ {2}}} . \ end {aligned}}} ### step 3

Especially for the real number

${\ displaystyle t_ {0} = {\ frac {\ operatorname {V} (X)} {c}}}$ applies after step 2:

{\ displaystyle {\ begin {aligned} \ operatorname {P} {{\ bigl (} X \ geq {\ operatorname {E} (X) + c} {\ bigr)}} & \ leq {\ frac {{\ operatorname {V} (X)} + {t_ {0}} ^ {2}} {(c + {t_ {0}}) ^ {2}}} \\ & = {\ frac {{\ operatorname {V} (X)} + {\ frac {{\ operatorname {V} (X)} ^ {2}} {c ^ {2}}}} {{(c + {\ frac {\ operatorname {V} (X)} {c}})} ^ {2}}} \\ & = {\ frac {{c ^ {2}} \ cdot {\ operatorname {V} (X)} + {{\ operatorname {V} (X) } ^ {2}}} {{{c ^ {2}} \ cdot (c + {\ frac {\ operatorname {V} (X)} {c}})} ^ {2}}} \\ & = { \ frac {{\ operatorname {V} (X)} \ cdot {{\ bigl (} {c ^ {2}} + {{\ operatorname {V} (X)} {\ bigr)}}}} {{ ({c ^ {2}} + {\ operatorname {V} (X)})} ^ {2}}} \\ & = {\ frac {\ operatorname {V} {{\ bigl (} X {\ bigr )}}} {c ^ {2} + {\ operatorname {V} (X)}}} \\\ end {aligned}}} .

So everything is proven.

## annotation

The real-valued function occurring in step 2 above

${\ displaystyle t \ mapsto {\ frac {{\ operatorname {V} (X)} + t ^ {2}} {(c + t) ^ {2}}} \; (t \ in \;] - c \;, \; \ infty [)}$ takes at the named place

${\ displaystyle t_ {0} = {\ frac {\ operatorname {V} (X)} {c}}}$ their bare minimum . The upper bound mentioned in the Cantellian inequality is therefore optimal in this sense.

## swell

• 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.${\ displaystyle \ xi}$ ${\ displaystyle \ operatorname {E} (\ xi)}$ 3. For a real random variable , its variance is designated.${\ displaystyle \ xi}$ ${\ displaystyle \ operatorname {V} (\ xi)}$ 