Noncentral chi-square distribution

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Noncentral chi-square
Probability density function
Cumulative distribution function
Parameters degrees of freedom

non-centrality parameter

Probability density function (pdf)
Cumulative distribution function (cdf) :
Excess kurtosis
Moment-generating function (mgf)
Characteristic function

In probability theory and statistics, the noncentral chi-square or noncentral distribution is a generalization of the chi-square distribution. If are k independent, normally distributed random variables with means and variances , then the random variable

is distributed according to the noncentral chi-square distribution. The noncentral chi-square distribution has two parameters: which specifies the number of degrees of freedom (i.e. the number of ), and which is related to the mean of the random variables by:

Note that some references define as one half of the above sum.


The noncentral chi-square distribution is equivalent to a (central) chi-square distribution with degrees of freedom, where is a Poisson random variable with parameter . Thus, the probability density function is given by

where is distributed as chi-square with degrees of freedom.

Alternatively, the pdf can be written as

where is a modified Bessel function of the first kind given by

The moment generating function is given by

The first few raw moments are:

The first few central moments are:

The nth cumulant is


Again using the relation between the central and noncentral chi-square distributions, the cumulative distribution function (cdf) can be written as

where is the cumulative density of the central chi-squared distribution which is given by

where is the lower incomplete Gamma function.

Derivation of the pdf

The derivation of the probability density function is most easily done by performing the following steps:

  1. Start with the joint PDF of two independent non-zero mean Gaussian distributions, and .
  2. Convert the joint density to polar: where , .
  3. Integrate over the angular variable .
  4. Convert from R to r where . This will yield a series expansion in r one factor of which matches the modified Bessel function .
  5. Take the Laplace (Fourier) transform term-by-term and the special case K = 2 and the MGF will result.
  6. For the general case, take the K = 2 MGF and raise it to the power.
  7. The final trick to hide the K-dependence in the numerator of the MGF is to note that is a function of K; that is,
and therefore, is not explicitly a function of K in the above table.

Related distributions

  • If is chi-square distributed then is also non-central chi-square distributed:
  • If , then
Various chi and chi-square distributions
Name Statistic
chi-square distribution
noncentral chi-square distribution
chi distribution
noncentral chi distribution


  • Abramowitz, M. and Stegun, I.A. (1972), Handbook of Mathematical Functions, Dover. Section 26.4.25.
  • Johnson, N. L. and Kotz, S., (1970), Continuous Univariate Distributions, vol. 2, Houghton-Mifflin.

Related Links

This distribution (and many others) is available in the free interactive statistical tables program, STATTAB. The cumulative distribution function, its inverse, and parameters of the distribution can be calculated with these packages. A free Fortran library for these distributions is in CDFLIB. The URL for download is [1]

it:Variabile casuale chi quadrato non centrale