# Difference between revisions of "Chi-squared distribution"

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− | + | '' $ \chi ^ {2} $- | |

+ | distribution'' | ||

− | + | The continuous probability distribution, concentrated on the positive semi-axis $ ( 0, \infty ) $, | |

+ | with density | ||

− | + | $$ | |

+ | p ( x) = | ||

+ | \frac{1}{2 ^ {n / 2 } \Gamma ( {n / 2 } ) } | ||

− | + | e ^ {- {x / 2 } } | |

+ | x ^ { {n / 2 } - 1 } , | ||

+ | $$ | ||

− | + | where $ \Gamma ( \alpha ) $ | |

+ | is the gamma-function and the positive integral parameter $ n $ | ||

+ | is called the number of degrees of freedom. A "chi-squared" distribution is a special case of a [[Gamma-distribution|gamma-distribution]] and has all the properties of the latter. The distribution function of a "chi-squared" distribution is an incomplete gamma-function, the characteristic function is expressed by the formula | ||

− | + | $$ | |

+ | \phi ( t) = \ | ||

+ | ( 1 - 2it) ^ {-} n/2 , | ||

+ | $$ | ||

− | + | and the mathematical expectation and variance are $ n $ | |

+ | and $ 2n $, | ||

+ | respectively. The family of "chi-squared" distributions is closed under the operation of convolution. | ||

− | + | The "chi-squared" distribution with $ n $ | |

+ | degrees of freedom can be derived as the distribution of the sum $ \chi _ {n} ^ {2} = X _ {1} ^ {2} + \dots + X _ {n} ^ {2} $ | ||

+ | of the squares of independent random variables $ X _ {1} \dots X _ {n} $ | ||

+ | having identical normal distributions with mathematical expectation 0 and variance 1. This connection with a normal distribution determines the role that the "chi-squared" distribution plays in probability theory and in mathematical statistics. | ||

− | + | Many distributions can be defined by means of the "chi-squared" distribution. For example, the distribution of the random variable $ \sqrt {\chi _ {n} ^ {2} } $— | |

+ | the length of the random vector $ ( X _ {1} \dots X _ {n} ) $ | ||

+ | with independent normally-distributed components — (sometimes called a "chi" -distribution, see also the special cases of a [[Maxwell distribution|Maxwell distribution]] and a [[Rayleigh distribution|Rayleigh distribution]]), the [[Student distribution|Student distribution]], and the [[Fisher-F-distribution|Fisher $ F $- | ||

+ | distribution]]. In mathematical statistics these distributions together with the "chi-squared" distribution describe sample distributions of various statistics of normally-distributed results of observations and are used to construct statistical interval estimators and statistical tests. A special reputation in connection with the "chi-squared" distribution has been gained by the [[Chi-squared test| "chi-squared" test]], based on the so-called "chi-squared" statistic of E.S. Pearson. | ||

+ | |||

+ | There are detailed tables of the "chi-squared" distribution which are convenient for statistical calculations. For large $ n $ | ||

+ | one uses approximations by means of a normal distribution; for example, according to the central limit theorem, the distribution of the normalized variable $ ( \chi _ {n} ^ {2} - n)/ \sqrt 2n $ | ||

+ | converges to the standard normal distribution. More accurate is the approximation | ||

+ | |||

+ | $$ | ||

+ | {\mathsf P} \{ \chi _ {n} ^ {2} < x \} \rightarrow \Phi ( \sqrt 2x - \sqrt {2n- 1 } ) \ \ | ||

+ | \textrm{ as } n \rightarrow \infty , | ||

+ | $$ | ||

+ | |||

+ | where $ \Phi ( x) $ | ||

+ | is the standard normal distribution function. | ||

See also [[Non-central chi-squared distribution|Non-central "chi-squared" distribution]]. | See also [[Non-central chi-squared distribution|Non-central "chi-squared" distribution]]. | ||

Line 25: | Line 64: | ||

====References==== | ====References==== | ||

<table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M.G. Kendall, A. Stuart, "The advanced theory of statistics. Distribution theory" , '''1''' , Griffin (1969)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> H.O. Lancaster, "The chi-squared distribution" , Wiley (1969)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> L.N. Bol'shev, N.V. Smirnov, "Tables of mathematical statistics" , ''Libr. math. tables'' , '''46''' , Nauka (1983) (In Russian) (Processed by L.S. Bark and E.S. Kedrova)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M.G. Kendall, A. Stuart, "The advanced theory of statistics. Distribution theory" , '''1''' , Griffin (1969)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> H.O. Lancaster, "The chi-squared distribution" , Wiley (1969)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> L.N. Bol'shev, N.V. Smirnov, "Tables of mathematical statistics" , ''Libr. math. tables'' , '''46''' , Nauka (1983) (In Russian) (Processed by L.S. Bark and E.S. Kedrova)</TD></TR></table> | ||

− | |||

− | |||

====Comments==== | ====Comments==== | ||

The name "chi-square" distribution is also used. | The name "chi-square" distribution is also used. |

## Latest revision as of 16:43, 4 June 2020

* $ \chi ^ {2} $-*
distribution

The continuous probability distribution, concentrated on the positive semi-axis $ ( 0, \infty ) $, with density

$$ p ( x) = \frac{1}{2 ^ {n / 2 } \Gamma ( {n / 2 } ) } e ^ {- {x / 2 } } x ^ { {n / 2 } - 1 } , $$

where $ \Gamma ( \alpha ) $ is the gamma-function and the positive integral parameter $ n $ is called the number of degrees of freedom. A "chi-squared" distribution is a special case of a gamma-distribution and has all the properties of the latter. The distribution function of a "chi-squared" distribution is an incomplete gamma-function, the characteristic function is expressed by the formula

$$ \phi ( t) = \ ( 1 - 2it) ^ {-} n/2 , $$

and the mathematical expectation and variance are $ n $ and $ 2n $, respectively. The family of "chi-squared" distributions is closed under the operation of convolution.

The "chi-squared" distribution with $ n $ degrees of freedom can be derived as the distribution of the sum $ \chi _ {n} ^ {2} = X _ {1} ^ {2} + \dots + X _ {n} ^ {2} $ of the squares of independent random variables $ X _ {1} \dots X _ {n} $ having identical normal distributions with mathematical expectation 0 and variance 1. This connection with a normal distribution determines the role that the "chi-squared" distribution plays in probability theory and in mathematical statistics.

Many distributions can be defined by means of the "chi-squared" distribution. For example, the distribution of the random variable $ \sqrt {\chi _ {n} ^ {2} } $— the length of the random vector $ ( X _ {1} \dots X _ {n} ) $ with independent normally-distributed components — (sometimes called a "chi" -distribution, see also the special cases of a Maxwell distribution and a Rayleigh distribution), the Student distribution, and the Fisher $ F $- distribution. In mathematical statistics these distributions together with the "chi-squared" distribution describe sample distributions of various statistics of normally-distributed results of observations and are used to construct statistical interval estimators and statistical tests. A special reputation in connection with the "chi-squared" distribution has been gained by the "chi-squared" test, based on the so-called "chi-squared" statistic of E.S. Pearson.

There are detailed tables of the "chi-squared" distribution which are convenient for statistical calculations. For large $ n $ one uses approximations by means of a normal distribution; for example, according to the central limit theorem, the distribution of the normalized variable $ ( \chi _ {n} ^ {2} - n)/ \sqrt 2n $ converges to the standard normal distribution. More accurate is the approximation

$$ {\mathsf P} \{ \chi _ {n} ^ {2} < x \} \rightarrow \Phi ( \sqrt 2x - \sqrt {2n- 1 } ) \ \ \textrm{ as } n \rightarrow \infty , $$

where $ \Phi ( x) $ is the standard normal distribution function.

See also Non-central "chi-squared" distribution.

#### References

[1] | H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946) |

[2] | M.G. Kendall, A. Stuart, "The advanced theory of statistics. Distribution theory" , 1 , Griffin (1969) |

[3] | H.O. Lancaster, "The chi-squared distribution" , Wiley (1969) |

[4] | L.N. Bol'shev, N.V. Smirnov, "Tables of mathematical statistics" , Libr. math. tables , 46 , Nauka (1983) (In Russian) (Processed by L.S. Bark and E.S. Kedrova) |

#### Comments

The name "chi-square" distribution is also used.

**How to Cite This Entry:**

Chi-squared distribution.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Chi-squared_distribution&oldid=18611