g01ef returns the lower or upper tail probability of the gamma distribution, with parameters

α

and

β

Syntax

C#
public static double g01ef( string tail, double g, double a, double b, out int ifail )

Visual Basic
Public Shared Function g01ef ( _ tail As String, _ g As Double, _ a As Double, _ b As Double, _ <OutAttribute> ByRef ifail As Integer _ ) As Double

Visual C++
public: static double g01ef( String^ tail, double g, double a, double b, [OutAttribute] int% ifail )

F#
static member g01ef : tail : string * g : float * a : float * b : float * ifail : int byref -> float

Parameters

tail

Type: System..::..String

On entry: indicates whether an upper or lower tail probability is required.

$tail = "L"$: The lower tail probability is returned, that is $P (G \leq g : α, β)$ .
$tail = "U"$: The upper tail probability is returned, that is $P (G \geq g : α, β)$ .

Constraint:

tail = "L"

"U"

g: Type: System..::..Double
On entry: $g$ , the value of the gamma variate.

Constraint: $g \geq 0.0$ .

a: Type: System..::..Double
On entry: the parameter $α$ of the gamma distribution.

Constraint: $a > 0.0$ .

b: Type: System..::..Double
On entry: the parameter $β$ of the gamma distribution.

Constraint: $b > 0.0$ .

ifail: Type: System..::..Int32%
On exit: $ifail = 0$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Return Value

g01ef returns the lower or upper tail probability of the gamma distribution, with parameters

α

and

β

Description

The lower tail probability for the gamma distribution with parameters

α

and

β

P (G \leq g)

, is defined by:

P (G \leq g; α, β) = \frac{1}{β^{α} Γ (α)} \int_{0}^{g} G^{α - 1} e^{- G / β} d G, α > 0.0, ​ β > 0.0 .

The mean of the distribution is

α β

and its variance is

α β^{2}

. The transformation

Z = \frac{G}{β}

is applied to yield the following incomplete gamma function in normalized form,

P (G \leq g; α, β) = P (Z \leq g / β : α, 1.0) = \frac{1}{Γ (α)} \int_{0}^{g / β} Z^{α - 1} e^{- Z} d Z .

This is then evaluated using s14ba.

References

Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth

Error Indicators and Warnings

Errors or warnings detected by the method:

ifail = 1

2

3

4

on exit, then g01ef returns

0.0

$ifail = 1$

On entry,

tail \neq "L"

"U"

$ifail = 2$

On entry,

g < 0.0

$ifail = 3$

On entry,	$a \leq 0.0$ ,
or	$b \leq 0.0$ .

$ifail = 4$: The solution did not converge in $600$ iterations. See s14ba. The probability returned should be a reasonable approximation to the solution.

$ifail = -9000$: An error occured, see message report.

Accuracy

The result should have a relative accuracy of machine precision. There are rare occasions when the relative accuracy attained is somewhat less than machine precision but the error should not exceed more than