# NAG FL Interfaceg01eaf (prob_​normal)

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## 1Purpose

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

## 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 <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.

## 3Description

The lower tail probability for the standard Normal distribution, $P\left(X\le x\right)$ is defined by:
 $P(X≤x)=∫-∞xZ(X)dX,$
where
 $Z(X)=12π e-X2/2, -∞
The relationship
 $P(X≤x)=12erfc(-x2)$
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
 $P(X≤|x|)-P(X≤-|x|)=erf(|x|2) ,$
is used, where erf is the error function, and is computed using s15aef.
NIST Digital Library of Mathematical Functions
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., $P\left(X\ge |x|\right)+P\left(X\le -|x|\right)$.
${\mathbf{tail}}=\text{'C'}$
The two tail (confidence interval) probability is returned, i.e., $P\left(X\le |x|\right)-P\left(X\le -|x|\right)$.
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: $\mathbf{ifail}$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 $-\mathbf{1}$ 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 $\text{'C'}$.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
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.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface 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)