Cdf.Gamma Method

Evaluates the gamma cumulative probability distribution function.

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

Parameters

x: A double specifying the argument at which the function is to be evaluated.
a: A double specifying the shape parameter. This must be positive.

Return Value

A double specifying the probability that a gamma random variable takes on a value less than or equal to x.

Remarks

Method 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 Gamma by setting .

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

Gamma Distribution Function

Cdf.Gamma Method

Parameters

Return Value

Remarks

See Also