In probability theory

Probability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

 and statistics

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, the noncentral chi distribution is a generalization of the chi distribution. If are k independent, normally distributed random variables with means and variances , then the statistic


is distributed according to the noncentral chi distribution. The noncentral chi distribution has two parameters: which specifies the number of degrees of freedom

Degrees of freedom (statistics)

In statistics, the number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary.Estimates of statistical parameters can be based upon different amounts of information or data. The number of independent pieces of information that go into the...

 (i.e. the number of ), and which is related to the mean of the random variables by:

Properties

The probability density function is


where is a modified Bessel function

Bessel function

In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y of Bessel's differential equation:...

 of the first kind.

The first few raw moments

Moment (mathematics)

In mathematics, a moment is, loosely speaking, a quantitative measure of the shape of a set of points. The "second moment", for example, is widely used and measures the "width" of a set of points in one dimension or in higher dimensions measures the shape of a cloud of points as it could be fit by...

 are:


where is the generalized Laguerre polynomial. Note that the 2nth moment is the same as the nth moment of the noncentral chi-squared distribution with being replaced by .

Related distributions

• If is a random variable with the non-central chi distribution, the random variable will have the noncentral chi-squared distribution. Other related distributions may be seen there.

• If is chi distributed: then is also non-central chi distributed: . In other words, the chi distribution is a special case of the non-central chi distribution (i.e., with a non-centrality parameter of zero).

• A noncentral chi distribution with 2 degrees of freedom is equivalent to a Rice distribution
  Rice distribution
  In probability theory, the Rice distribution or Rician distribution is the probability distribution of the absolute value of a circular bivariate normal random variable with potentially non-zero mean. It was named after Stephen O...
  
   with .

• If X follows a noncentral chi distribution with 1 degree of freedom and noncentrality parameter λ, then σX follows a folded normal distribution
  Folded Normal Distribution
  The folded normal distribution is a probability distribution related to the normal distribution. Given a normally distributed random variable X with mean μ and variance σ2, the random variable Y = |X| has a folded normal distribution. Such a case may be encountered if only the magnitude of some...
  
  whose parameters are equal to σλ and σ² for any value of σ.