NAG FL Interface
g01eaf (prob_​normal)

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1 Purpose

g01eaf returns a one or two tail probability for the standard Normal distribution.

2 Specification

Fortran Interface
Function g01eaf ( tail, x, ifail)
Real (Kind=nag_wp) :: g01eaf
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: x
Character (1), Intent (In) :: tail
C Header Interface
#include <nag.h>
double  g01eaf_ (const char *tail, const double *x, Integer *ifail, const Charlen length_tail)
The routine may be called by the names g01eaf or nagf_stat_prob_normal.

3 Description

The lower tail probability for the standard Normal distribution, P(Xx) is defined by:
P(Xx)=-xZ(X)dX,  
where
Z(X)=12π e-X2/2, -<X< .  
The relationship
P(Xx)=12erfc(-x2)  
is used, where erfc is the complementary error function, and is computed using s15adf. For the upper tail probability the relationship P(Xx)=P(X-x) is used and for the two tail significance level probability twice the probability obtained from the absolute value of x is returned.
When the two tail confidence probability is required the relationship
P(X|x|)-P(X-|x|)=erf(|x|2) ,  
is used, where erf is the error function, and is computed using s15aef.

4 References

NIST Digital Library of Mathematical Functions
Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth

5 Arguments

1: tail Character(1) Input
On entry: indicates which tail the returned probability should represent.
tail='L'
The lower tail probability is returned, i.e., P(Xx).
tail='U'
The upper tail probability is returned, i.e., P(Xx).
tail='S'
The two tail (significance level) probability is returned, i.e., P(X|x|)+P(X-|x|).
tail='C'
The two tail (confidence interval) probability is returned, i.e., P(X|x|)-P(X-|x|).
Constraint: tail='L', 'U', 'S' or 'C'.
2: x Real (Kind=nag_wp) Input
On entry: x, the value of the standard Normal variate.
3: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry ifail=0 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
If ifail0, then g01eaf returns 0.0.
ifail=1
On entry, tail=value.
Constraint: tail='L', 'U', 'S' or 'C'.
ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
ifail=-399
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
ifail=-999
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

Accuracy is limited by machine precision. For detailed error analysis see s15adf and s15aef.

8 Parallelism and Performance

g01eaf is not threaded in any implementation.

9 Further Comments

None.

10 Example

Four values of tail and x are input and the probabilities calculated and printed.

10.1 Program Text

Program Text (g01eafe.f90)

10.2 Program Data

Program Data (g01eafe.d)

10.3 Program Results

Program Results (g01eafe.r)