# Non-central chi-squared distribution

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. The 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 and variance (cf. 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).

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. 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

 [1] 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) [2] M.G. Kendall, A. Stuart, "The advanced theory of statistics" , 2. Inference and relationship , Griffin (1979) [3] P.B. Patnaik, "The non-central $\chi ^ { 2 }$- and $F$-distributions and their applications" Biometrica , 36 (1949) pp. 202–232