View source for Non-central chi-squared distribution

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''non-central  $  \chi  ^ {2} $-
distribution''

A continuous probability distribution concentrated on the positive semi-axis  $  0 < x < \infty $
with density

$$ 

\frac{e ^ {- ( x + \lambda ) / 2 } x ^ {( n - 2 ) / 2 } }{2 ^ {n / 2 } \Gamma ( 1 / 2 ) }

\sum _ {r = 0 } ^  \infty  

\frac{\lambda  ^ {r} x  ^ {r} }{( 2 r ) ! }

\frac{\Gamma ( r + 1 / 2 ) }{\Gamma ( r + n / 2 ) }
 ,
$$

where  $  n $
is the number of degrees of freedom and  $  \lambda $
the parameter of non-centrality. For  $  \lambda = 0 $
this density is that of the ordinary (central) [[Chi-squared distribution| "chi-squared"  distribution]]. The [[Characteristic function|characteristic function]] of a non-central  "chi-squared"  distribution is

$$ 
\phi ( t)  =  ( 1 - 2 i t )  ^ {-} n/2  \mathop{\rm exp}
\left \{ 
\frac{\lambda i t }{1 - 2 i t }
 \right \} ;
$$

the [[Mathematical expectation|mathematical expectation]] and variance (cf. [[Dispersion|Dispersion]]) are  $  n + \lambda $
and  $  2 ( n + 2 \lambda ) $, 
respectively. A non-central  "chi-squared"  distribution belongs to the class of infinitely-divisible distributions (cf. [[Infinitely-divisible distribution|Infinitely-divisible distribution]]).

As a rule, a non-central  "chi-squared"  distribution appears as the distribution of the sum of squares of independent random variables  $  X _ {1} \dots X _ {n} $
having normal distributions with non-zero means  $  m _ {i} $
and unit variance; more precisely, the sum  $  X _ {1}  ^ {2} + \dots X _ {n}  ^ {2} $
has a non-central  "chi-squared"  distribution with  $  n $
degrees of freedom and non-centrality parameter  $  \lambda = \sum _ {i=} 1  ^ {n} m _ {i}  ^ {2} $. 
The sum of several mutually independent random variables with a non-central  "chi-squared"  distribution has a distribution of the same type and its parameters are the sums of the corresponding parameters of the summands.

If  $  n $
is even, then the distribution function of a non-central  "chi-squared"  distribution  $  F _ {n} ( x ;  \lambda ) $
is given by  $  F _ {n} ( x ;  \lambda ) = 0 $
for  $  x \leq  0 $
and for  $  x > 0 $
by

$$ 
F _ {n} ( x ;  \lambda )  = \ 
\sum _ { m= } 0 ^  \infty   \ 
\sum _ {k = m + n / 2 } ^  \infty  

\frac{( \lambda / 2 )  ^ {m} ( x / 2 )  ^ {k} }{m ! k ! }

e ^ {- ( \lambda + x ) / 2 } .
$$

This formula establishes a link between a non-central  "chi-squared"  distribution and a [[Poisson distribution|Poisson distribution]]. Namely, if  $  X $
and  $  Y $
have Poisson distributions with parameters  $  x / 2 $
and  $  \lambda / 2 $, 
respectively, then for any positive integer  $  s > 0 $,

$$ 
{\mathsf P} \{ X - Y \geq  s \}  =  F _ {2s} ( x ;  \lambda ) .
$$

A non-central  "chi-squared"  distribution often arises in problems of mathematical statistics concerned with the study of the power of tests of  "chi-squared"  type. Since tables of non-central  "chi-squared"  distributions are fairly complete, various approximations by means of a  "chi-squared"  and a normal distribution are widely used in statistical applications.

====References====
<table><TR><TD valign="top">[1]</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><TR><TD valign="top">[2]</TD> <TD valign="top">  M.G. Kendall,   A. Stuart,   "The advanced theory of statistics" , '''2. Inference and relationship''' , Griffin  (1979)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P.B. Patnaik,   "The non-central <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696028.png" />- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696029.png" />-distributions and their applications"  ''Biometrica'' , '''36'''  (1949)  pp. 202–232</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.L. Johnson,   S. Kotz,   "Distributions in statistics" , '''2. Continuous univariate distributions''' , Wiley  (1970)</TD></TR></table>