# proof of Chebyshev’s inequality

The proof of Chebyshev’s inequality follows from the application of Markov’s inequality.

Define $Y={(X-\mu )}^{2}$. Then $Y\ge 0$ is a random variable^{}, and

$$\mathbb{E}[Y]=\mathrm{Var}[X]={\sigma}^{2}.$$ |

Applying Markov’s inequality to $Y$, we see that

$$\mathbb{P}\left\{\right|X-\mu |\ge t\}=\mathbb{P}\{Y\ge {t}^{2}\}\le \frac{1}{{t}^{2}}\mathbb{E}[Y]=\frac{{\sigma}^{2}}{{t}^{2}}.$$ |

Title | proof of Chebyshev’s inequality |
---|---|

Canonical name | ProofOfChebyshevsInequality |

Date of creation | 2013-03-22 12:47:58 |

Last modified on | 2013-03-22 12:47:58 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 6 |

Author | PrimeFan (13766) |

Entry type | Proof |

Classification | msc 60A99 |