non-central chi-squared random variable
is called a non-central chi-squared random variable. Its distribution depends only on the number of degrees of freedom and non-centrality parameter . This is denoted by and has moment generating function
which is defined for all with real part less than . More generally, for any , not necessarily integers, a random variable has the non-central chi-squared distribution, , if its moment generating function is given by (1).
A non-central chi-squared random variable for any can be constructed as follows. Let be a (central) chi-squared variable with degree , be standard normals, and have the distribution. If these are all independent then
has the distribution. Correspondingly, the probability density function for is
where and is the probability density of the distribution. Alternatively, this can be expressed as
where is a modified Bessel function of the first kind,
has mean and variance .
. The (central) chi-squared random variable is a special case of the non-central chi-squared random variable, when the non-centrality parameter .
If then the distribution is restricted to the domain with probability density function (2). On the other hand, if , then there is also an atom at ,
If is a multivariate normally distributed -dimensional random vector with distribution where is the mean vector and is the covariance matrix. Suppose that is singular, with = rank of . Then is a non-central chi-squared random variable, where is a generalized inverse of . Its distribution has degrees of freedom with non-centrality parameter .
|Title||non-central chi-squared random variable|
|Date of creation||2013-03-22 14:56:16|
|Last modified on||2013-03-22 14:56:16|
|Last modified by||CWoo (3771)|
|Synonym||non-central chi-squared distribution|