# NAG Library Routine Document

## 1Purpose

g01eaf returns a one or two tail probability for the standard Normal distribution, via the routine name.

## 2Specification

Fortran Interface
 Function g01eaf ( tail, x,
 Real (Kind=nag_wp) :: g01eaf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x Character (1), Intent (In) :: tail
#include nagmk26.h
 double g01eaf_ (const char *tail, const double *x, Integer *ifail, const Charlen length_tail)

## 3Description

The lower tail probability for the standard Normal distribution, $P\left(X\le x\right)$ is defined by:
 $PX≤x=∫-∞xZXdX,$
where
 $ZX=12π e-X2/2, -∞
The relationship
 $PX≤x=12erfc-x2$
is used, where erfc is the complementary error function, and is computed using s15adf. For the upper tail probability the relationship $P\left(X\ge x\right)=P\left(X\le -x\right)$ 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
 $PX≤x-PX≤-x=erfx2 ,$
is used, where erf is the error function, and is computed using s15aef.

## 4References

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

## 5Arguments

1:     $\mathbf{tail}$ – Character(1)Input
On entry: indicates which tail the returned probability should represent.
${\mathbf{tail}}=\text{'L'}$
The lower tail probability is returned, i.e., $P\left(X\le x\right)$.
${\mathbf{tail}}=\text{'U'}$
The upper tail probability is returned, i.e., $P\left(X\ge x\right)$.
${\mathbf{tail}}=\text{'S'}$
The two tail (significance level) probability is returned, i.e., $P\left(X\ge \left|x\right|\right)+P\left(X\le -\left|x\right|\right)$.
${\mathbf{tail}}=\text{'C'}$
The two tail (confidence interval) probability is returned, i.e., $P\left(X\le \left|x\right|\right)-P\left(X\le -\left|x\right|\right)$.
Constraint: ${\mathbf{tail}}=\text{'L'}$, $\text{'U'}$, $\text{'S'}$ or $\text{'C'}$.
2:     $\mathbf{x}$ – Real (Kind=nag_wp)Input
On entry: $x$, the value of the standard Normal variate.
3:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ 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 ${\mathbf{ifail}}\ne {\mathbf{0}}$, then g01eaf returns $0.0$.
${\mathbf{ifail}}=1$
On entry, ${\mathbf{tail}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{tail}}=\text{'L'}$, $\text{'U'}$, $\text{'S'}$ or $\text{'C'}$.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

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

## 8Parallelism and Performance

g01eaf is not threaded in any implementation.

None.

## 10Example

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

### 10.1Program Text

Program Text (g01eafe.f90)

### 10.2Program Data

Program Data (g01eafe.d)

### 10.3Program Results

Program Results (g01eafe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017