g01bl returns the lower tail, upper tail and point probabilities associated with a hypergeometric distribution.

Syntax

C#
public static void g01bl(
	int n,
	int l,
	int m,
	int k,
	out double plek,
	out double pgtk,
	out double peqk,
	out int ifail
)
Visual Basic
Public Shared Sub g01bl ( _
	n As Integer, _
	l As Integer, _
	m As Integer, _
	k As Integer, _
	<OutAttribute> ByRef plek As Double, _
	<OutAttribute> ByRef pgtk As Double, _
	<OutAttribute> ByRef peqk As Double, _
	<OutAttribute> ByRef ifail As Integer _
)
Visual C++
public:
static void g01bl(
	int n, 
	int l, 
	int m, 
	int k, 
	[OutAttribute] double% plek, 
	[OutAttribute] double% pgtk, 
	[OutAttribute] double% peqk, 
	[OutAttribute] int% ifail
)
F#
static member g01bl : 
        n : int * 
        l : int * 
        m : int * 
        k : int * 
        plek : float byref * 
        pgtk : float byref * 
        peqk : float byref * 
        ifail : int byref -> unit

Parameters

n
Type: System..::..Int32
On entry: the parameter n of the hypergeometric distribution.
Constraint: n≥0.
l
Type: System..::..Int32
On entry: the parameter l of the hypergeometric distribution.
Constraint: 0≤l≤n.
m
Type: System..::..Int32
On entry: the parameter m of the hypergeometric distribution.
Constraint: 0≤m≤n.
k
Type: System..::..Int32
On entry: the integer k which defines the required probabilities.
Constraint: max0,l-n-m≤k≤minl,m.
plek
Type: System..::..Double%
On exit: the lower tail probability, ProbX≤k.
pgtk
Type: System..::..Double%
On exit: the upper tail probability, ProbX>k.
peqk
Type: System..::..Double%
On exit: the point probability, ProbX=k.
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]).

Description

Let X denote a random variable having a hypergeometric distribution with parameters n, l and m (n≥l≥0, n≥m≥0). Then
ProbX=k=mkn-ml-knl,
where max0,l-n-m≤k≤minl,m, 0≤l≤n and 0≤m≤n.
The hypergeometric distribution may arise if in a population of size n a number m are marked. From this population a sample of size l is drawn and of these k are observed to be marked.
The mean of the distribution =lmn, and the variance =lmn-ln-mn2n-1.
g01bl computes for given n, l, m and k the probabilities:
plek=ProbX≤kpgtk=ProbX>kpeqk=ProbX=k.
The method is similar to the method for the Poisson distribution described in Knüsel (1986).

References

Knüsel L (1986) Computation of the chi-square and Poisson distribution SIAM J. Sci. Statist. Comput. 7 1022–1036

Error Indicators and Warnings

Errors or warnings detected by the method:
ifail=1
On entry,n<0.
ifail=2
On entry,l<0,
orl>n.
ifail=3
On entry,m<0,
orm>n.
ifail=4
On entry,k<0,
ork>l,
ork>m,
ork<l+m-n.
ifail=5
On entry,n is too large to be represented exactly as a real number.
ifail=6
On entry,the variance (see [Description]) exceeds 106.
ifail=-9000
An error occured, see message report.

Accuracy

Results are correct to a relative accuracy of at least 10-6 on machines with a precision of 9 or more decimal digits, and to a relative accuracy of at least 10-3 on machines of lower precision (provided that the results do not underflow to zero).

Parallelism and Performance

None.

Further Comments

The time taken by g01bl depends on the variance (see [Description]) and on k. For given variance, the time is greatest when k≈lm/n (= the mean), and is then approximately proportional to the square-root of the variance.

Example

This example reads values of n, l, m and k from a data file until end-of-file is reached, and prints the corresponding probabilities.

Example program (C#): g01ble.cs

Example program data: g01ble.d

Example program results: g01ble.r

See Also