Evaluates the noncentral beta probability density function.

Namespace: Imsl.Stat
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.0.0

Syntax

C#
public static double NoncentralBeta(
	double x,
	double shape1,
	double shape2,
	double lambda
)
Visual Basic (Declaration)
Public Shared Function NoncentralBeta ( _
	x As Double, _
	shape1 As Double, _
	shape2 As Double, _
	lambda As Double _
) As Double
Visual C++
public:
static double NoncentralBeta(
	double x, 
	double shape1, 
	double shape2, 
	double lambda
)

Parameters

x
Type: System..::.Double
A double scalar value representing the argument at which the function is to be evaluated. x must be nonnegative and less than or equal to 1.
shape1
Type: System..::.Double
A double scalar value representing the first shape parameter. shape1 must be positive.
shape2
Type: System..::.Double
A double scalar value representing the second shape parameter. shape2 must be positive.
lambda
Type: System..::.Double
A double scalar value representing the noncentrality parameter. lambda must nonnegative.

Return Value

A double scalar value representing the probability density associated with a noncentral beta random variable with value x.

Remarks

The noncentral beta distribution is a generalization of the beta distribution. If Z is a noncentral chi-square random variable with noncentrality parameter \lambda and 2 \alpha_1 degrees of freedom, and Y is a chi-square random variable with 2 \alpha_2 degrees of freedom which is statistically independent of Z, then

X \;\; = \;\; \frac{Z}{Z \; + \; Y} \;\; = \;\; \frac{\alpha_1 F}{\alpha_1 F \; + \; \alpha_2}

is a noncentral beta-distributed random variable and

F \;\; = \;\; \frac{\alpha_2 Z}{\alpha_1 Y} \;\; = \;\; \frac{\alpha_2 X}{\alpha_1 (1 \; - \; X)}

is a noncentral F-distributed random variable. The PDF for noncentral beta variable X can thus be simply defined in terms of the noncentral F PDF:

PDF_{nc\beta}(x, \; \alpha_1, \; \alpha_2, \; \lambda) \;\; = \;\; PDF_{ncF}(f, \; 2 \alpha_1, \; 2 \alpha_2, \; \lambda) \; \frac{df}{dx}

where PDF_{nc\beta}(x, \; \alpha_1, \; \alpha_2, \; \lambda) is the noncentral beta PDF with x = x, \alpha_1 = shape1, \alpha_2 = shape2, and noncentrality parameter \lambda = lambda; PDF_{ncF}(f, \; 2 \alpha_1, \; 2 \alpha_2, \; \lambda) is the noncentral F PDF with argument f, numerator and denominator degrees of freedom 2 \alpha_1 and 2 \alpha_2 respectively, noncentrality parameter \lambda,

f \;\; = \;\; \frac{\alpha_2 x}{\alpha_1 (1 \; - \; x)},
x \;\; = \;\; \frac{\alpha_1 f}{\alpha_1 f \; + \; \alpha_2},
and
\frac{df}{dx} \;\; = \;\; \frac{(\alpha_1 f \; + \; \alpha_2)^2}{\alpha_1 \alpha_2} \;\; = \;\; \frac{\alpha_2}{\alpha_1 (1 \; - \; x)^2}.

(See documentation for class Pdf method NoncentralF for a discussion of how the noncentral F PDF is defined and calculated.)

With a noncentrality parameter of zero, the noncentral beta distribution is the same as the beta distribution.

See Also