# Chi distribution

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Probability density function | |

Cumulative distribution function | |

Parameters
| (degrees of freedom) |

Support
| |

pdf
| |

cdf
| |

Mean
| |

Median
| |

Mode
| for |

Variance
| |

Skewness
| |

Kurtosis
| |

Entropy
| |

mgf
| Complicated (see text) |

Char. func.
| Complicated (see text) |

In probability theory and statistics, the **chi distribution** is a continuous probability distribution. The distribution usually arises when a k-dimensional vector's orthogonal components are independent and each follow a standard normal distribution. The length of the vector will then have a chi distribution. The most familiar example is the Maxwell distribution of (normalized) molecular speeds which is a chi distribution with 3 degrees of freedom. If are *k* independent, normally distributed random variables with means and standard deviations , then the statistic

is distributed according to the chi distribution. The chi distribution has one parameter: which specifies the number of degrees of freedom (i.e. the number of ).

## Properties Edit

The probability density function is

where is the Gamma function. The cumulative distribution function is given by:

where is the regularized Gamma function. The moment generating function is given by:

where is Kummer's confluent hypergeometric function. The raw moments are then given by:

where is the Gamma function. The first few raw moments are:

where the rightmost expressions are derived using the recurrence relationship for the Gamma function:

From these expressions we may derive the following relationships:

Mean:

Variance:

Skewness:

Kurtosis excess:

The characteristic function is given by:

where again, is Kummer's confluent hypergeometric function. The entropy is given by:

where is the polygamma function.

## Related distributionsEdit

- If is chi distributed then is chi-square distributed:
- The Rayleigh distribution with is a chi distribution with two degrees of freedom.
- The Maxwell distribution for normalized molecular speeds is a chi distribution with three degrees of freedom.
- The chi distribution for is the half-normal distribution.

Name | Statistic |
---|---|

chi-square distribution | |

noncentral chi-square distribution | |

chi distribution | |

noncentral chi distribution |

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