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m (moved "Chi-squared" distribution to Chi-squared distribution: Modify alphabetical position so this item will appear in "C" rather than before "A".)
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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022100/c0221002.png" />-distribution''
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The continuous probability distribution, concentrated on the positive semi-axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022100/c0221003.png" />, with density
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{{TEX|auto}}
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022100/c0221004.png" /></td> </tr></table>
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'' $  \chi  ^ {2} $-
 +
distribution''
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022100/c0221005.png" /> is the gamma-function and the positive integral parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022100/c0221006.png" /> 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
+
The continuous probability distribution, concentrated on the positive semi-axis $ ( 0, \infty ) $,  
 +
with density
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022100/c0221008.png" /></td> </tr></table>
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$$
 +
p ( x)  =
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\frac{1}{2 ^ {n / 2 } \Gamma ( {n / 2 } ) }
  
and the mathematical expectation and variance are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022100/c0221009.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022100/c02210010.png" />, respectively. The family of  "chi-squared"  distributions is closed under the operation of convolution.
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e ^ {- {x / 2 } }
 +
x ^ { {n / 2 } - 1 } ,
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$$
  
The "chi-squared"  distribution with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022100/c02210011.png" /> degrees of freedom can be derived as the distribution of the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022100/c02210012.png" /> of the squares of independent random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022100/c02210013.png" /> 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.
+
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
  
Many distributions can be defined by means of the  "chi-squared" distribution. For example, the distribution of the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022100/c02210014.png" /> — the length of the random vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022100/c02210015.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022100/c02210016.png" />-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.
+
$$
 +
\phi ( t) = \
 +
( 1 - 2it) ^ {-} n/2 ,
 +
$$
  
There are detailed tables of the "chi-squared"  distribution which are convenient for statistical calculations. For large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022100/c02210017.png" /> one uses approximations by means of a normal distribution; for example, according to the central limit theorem, the distribution of the normalized variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022100/c02210018.png" /> converges to the standard normal distribution. More accurate is the approximation
+
and the mathematical expectation and variance are $  n $
 +
and  $  2n $,
 +
respectively. The family of  "chi-squared"  distributions is closed under the operation of convolution.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022100/c02210019.png" /></td> </tr></table>
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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.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022100/c02210020.png" /> is the standard normal distribution function.
+
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]].
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====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=19334
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article