# covariance

The *covariance ^{}* of two random variables

^{}${X}_{1}$ and ${X}_{2}$ with mean (http://planetmath.org/ExpectedValue) ${\mu}_{1}$ and ${\mu}_{2}$ respectively is defined as

$$\mathrm{cov}({X}_{1},{X}_{2}):=E[({X}_{1}-{\mu}_{1})({X}_{2}-{\mu}_{2})].$$ | (1) |

The covariance of a random variable $X$ with itself is simply the variance^{}, $E[{(X-\mu )}^{2}]$.

Covariance captures a measure of the correlation^{} of two variables. Positive covariance indicates that as ${X}_{1}$ increases, so does ${X}_{2}$. Negative covariance indicates ${X}_{1}$ decreases as ${X}_{2}$ increases and vice versa. Zero covariance can indicate that ${X}_{1}$ and ${X}_{2}$ are uncorrelated.

The *correlation coefficient* provides a normalized view of correlation based on covariance:

$$\mathrm{corr}(X,Y):=\frac{\mathrm{cov}(X,Y)}{\sqrt{\mathrm{var}(X)\mathrm{var}(Y)}}.$$ | (2) |

$\mathrm{corr}(X,Y)$ ranges from -1 (for negatively correlated variables) through zero (for uncorrelated variables) to +1 (for positively correlated variables).

While if $X$ and $Y$ are independent^{} we have $\mathrm{corr}(X,Y)=0$, the latter does not imply the former.

Title | covariance |
---|---|

Canonical name | Covariance |

Date of creation | 2013-03-22 12:19:29 |

Last modified on | 2013-03-22 12:19:29 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 9 |

Author | Koro (127) |

Entry type | Definition |

Classification | msc 62-00 |

Synonym | cov |

Synonym | correlation |

Synonym | correlation coefficient |

Related topic | Variance |