Cdf.Gamma Method

CdfGamma Method

Evaluates the gamma cumulative probability distribution function.

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

Syntax

public static double Gamma(
	double x,
	double a
)

Public Shared Function Gamma ( 
	x As Double,
	a As Double
) As Double

public:
static double Gamma(
	double x, 
	double a
)

static member Gamma : 
        x : float * 
        a : float -> float

Parameters

x: Type: SystemDouble
A double scalar value representing the argument at which the function is to be evaluated.
a: Type: SystemDouble
A double scalar value representing the shape parameter. This must be positive.

Return Value

Type: Double
A double scalar value representing the probability that a gamma random variable takes on a value less than or equal to x.

Remarks

Method Cdf.Gamma evaluates the distribution function, F, of a gamma random variable with shape parameter a; that is,

$F\left(x\right)=\frac{1}{{\Gamma\left(a \right)}}\int_0^x{e^{-t}t^{a-1}}dt$

where $\Gamma(\cdot)$ is the gamma function. (The gamma function is the integral from 0 to $\infty$ of the same integrand as above). The value of the distribution function at the point x is the probability that the random variable takes a value less than or equal to x.

The gamma distribution is often defined as a two-parameter distribution with a scale parameter b (which must be positive), or even as a three-parameter distribution in which the third parameter c is a location parameter. In the most general case, the probability density function over $(c, \infty)$ is

$f\left(t\right)=\frac{1}{{b^a\Gamma \left(a\right)}}e^{-\left({t-c}\right)/b}\left({x-c}\right)^{a-1}$

If T is such a random variable with parameters a, b, and c, the probability that $T\le t_0$ can be obtained from Cdf.Gamma by setting .

If X is less than a or if X is less than or equal to 1.0, Cdf.Gamma uses a series expansion. Otherwise, a continued fraction expansion is used. (See Abramowitz and Stegun, 1964.)

Gamma Distribution Function

Reference

Cdf Class

Imsl.Stat Namespace

Other Resources

Example