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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]].

====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>

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