View source for Non-central chi-squared distribution

''non-central <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n0669602.png" />-distribution''

A continuous probability distribution concentrated on the positive semi-axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n0669603.png" /> 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/n/n066/n066960/n0669604.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n0669605.png" /> is the number of degrees of freedom and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n0669606.png" /> the parameter of non-centrality. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n0669607.png" /> 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

<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/n/n066/n066960/n0669608.png" /></td> </tr></table>

the [[Mathematical expectation|mathematical expectation]] and variance (cf. [[Dispersion|Dispersion]]) are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n0669609.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696010.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696011.png" /> having normal distributions with non-zero means <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696012.png" /> and unit variance; more precisely, the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696013.png" /> has a non-central  "chi-squared"  distribution with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696014.png" /> degrees of freedom and non-centrality parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696015.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696016.png" /> is even, then the distribution function of a non-central  "chi-squared"  distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696017.png" /> is given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696018.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696019.png" /> and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696020.png" /> by

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This formula establishes a link between a non-central  "chi-squared"  distribution and a [[Poisson distribution|Poisson distribution]]. Namely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696023.png" /> have Poisson distributions with parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696025.png" />, respectively, then for any positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696026.png" />,

<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/n/n066/n066960/n06696027.png" /></td> </tr></table>

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>