g01ec returns the lower or upper tail probability for the

χ^{2}

-distribution with real degrees of freedom.

Syntax

C#
public static double g01ec( string tail, double x, double df, out int ifail )

Visual Basic
Public Shared Function g01ec ( _ tail As String, _ x As Double, _ df As Double, _ <OutAttribute> ByRef ifail As Integer _ ) As Double

Visual C++
public: static double g01ec( String^ tail, double x, double df, [OutAttribute] int% ifail )

F#
static member g01ec : tail : string * x : float * df : float * ifail : int byref -> float

tail

Type: System..::..String

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

$tail = "L"$: The lower tail probability is returned, i.e., $P (X \leq x : ν)$ .
$tail = "U"$: The upper tail probability is returned, i.e., $P (X \geq x : ν)$ .

Constraint:

tail = "L"

"U"

x: Type: System..::..Double
On entry: $x$ , the value of the $χ^{2}$ variate with $ν$ degrees of freedom.

Constraint: $x \geq 0.0$ .

df: Type: System..::..Double
On entry: $ν$ , the degrees of freedom of the $χ^{2}$ -distribution.

Constraint: $df > 0.0$ .

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]).

g01ec returns the lower or upper tail probability for the

χ^{2}

-distribution with real degrees of freedom.

Description

The lower tail probability for the

χ^{2}

-distribution with

ν

degrees of freedom,

P (X \leq x : ν)

is defined by:

P (X \leq x : ν) = \frac{1}{2^{ν / 2} Γ (ν / 2)} \int_{0.0}^{x} X^{ν / 2 - 1} e^{- X / 2} d X, x \geq 0, ν > 0 .

To calculate

P (X \leq x : ν)

a transformation of a gamma distribution is employed, i.e., a

χ^{2}

-distribution with

ν

degrees of freedom is equal to a gamma distribution with scale parameter

2

and shape parameter

ν / 2

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

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

Note: g01ec may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the method:

ifail = 1

2

3

on exit, then g01ec returns

0.0

$ifail = 1$

On entry,

tail \neq "L"

"U"

$ifail = 2$

On entry,

x < 0.0

$ifail = 3$

On entry,

df \leq 0.0

$ifail = 4$: The solution has failed to converge while calculating the gamma variate. The result returned should represent an approximation to the solution.