Pdf.NoncentralChi Method

Evaluates the noncentral chi-squared probability density function.

Namespace: Imsl.Stat
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.2.0

Syntax

public static double NoncentralChi(
	double chsq,
	double df,
	double alam
)

Public Shared Function NoncentralChi ( 
	chsq As Double,
	df As Double,
	alam As Double
) As Double

public:
static double NoncentralChi(
	double chsq, 
	double df, 
	double alam
)

static member NoncentralChi : 
        chsq : float * 
        df : float * 
        alam : float -> float

Parameters

chsq: Type: System.Double
A double scalar value at which the function is to be evaluated. chsq must be nonnegative.
df: Type: System.Double
A double scalar value representing the number of degrees of freedom. df must be positive.
alam: Type: System.Double
A double scalar value representing the noncentrality parameter. alam must be nonnegative.

Return Value

Type: Double
A double scalar value representing the probability density associated with a noncentral chi-squared random variable with value chsq.

Remarks

The noncentral chi-squared distribution is a generalization of the chi-squared distribution. If $\{X_i\}$ are independent, normally distributed random variables with means $\mu_i$ and variances $\sigma^2_i$ , then the random variable

$X \;\; = \;\; \sum_{i = 1}^k \left(\frac{X_i}{\sigma_i}\right)^2$

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

$\lambda \;\; = \;\; \sum_{i = 1}^k \left(\frac{\mu_i}{\sigma_i}\right)^2$

The noncentral chi-squared distribution is equivalent to a (central) chi-squared distribution with degrees of freedom, where is the value of a Poisson distributed random variable with parameter $\lambda/2$ . Thus, the probability density function is given by:

$F(x,k,\lambda) \;\; = \;\; \sum_{i = 0}^\infty {\frac{e^{-\lambda/2} (\lambda/2)^i}{i!}} f(x,k+2i)$

where the (central) chi-squared Pdf is given by:

$f(x, k) \;\; = \;\; \frac{(x/2)^{k/2} \; e^{-x/2}}{x \; \Gamma(k/2)} \quad for \;\; x \; > \; 0, \;\; else \;\; 0$

where $\Gamma (\cdot)$ is the gamma function. The above representation of $F(x,k,\lambda)$ can be shown to be equivalent to the representation:

$F(x,k,\lambda) \;\; = \;\; \frac{e^{-(\lambda+x)/2} \; (x/2)^{k/2}}{x} \; \sum_{i = 0}^\infty {\phi_i}$

$\phi_i \;\; = \;\; \frac{(\lambda x / 4)^i}{i! \; \Gamma(k/2 \;\; + \;\; i)}$

Method Pdf.NoncentralChi evaluates the probability density function, $F(x,k,\lambda)$ , of a noncentral chi-squared random variable with df degrees of freedom and noncentrality parameter alam, corresponding to k = df, $\lambda$ = alam, and x = chsq.

With a noncentrality parameter of zero, the noncentral chi-squared distribution is the same as the central chi-squared distribution.

Reference

Pdf Class

Imsl.Stat Namespace