# NAG FL Interfaceg01saf (prob_​normal_​vector)

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## 1Purpose

g01saf returns a number of one or two tail probabilities for the Normal distribution.

## 2Specification

Fortran Interface
 Subroutine g01saf ( tail, lx, x, lxmu, xmu, xstd, p,
 Integer, Intent (In) :: ltail, lx, lxmu, lxstd Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: ivalid(*) Real (Kind=nag_wp), Intent (In) :: x(lx), xmu(lxmu), xstd(lxstd) Real (Kind=nag_wp), Intent (Out) :: p(*) Character (1), Intent (In) :: tail(ltail)
#include <nag.h>
 void g01saf_ (const Integer *ltail, const char tail[], const Integer *lx, const double x[], const Integer *lxmu, const double xmu[], const Integer *lxstd, const double xstd[], double p[], Integer ivalid[], Integer *ifail, const Charlen length_tail)
The routine may be called by the names g01saf or nagf_stat_prob_normal_vector.

## 3Description

The lower tail probability for the Normal distribution, $P\left({X}_{i}\le {x}_{i}\right)$ is defined by:
 $P(Xi≤xi) = ∫ -∞ xi Zi(Xi)dXi ,$
where
 $Zi(Xi) = 1 2πσi2 e -(Xi-μi)2/(2σi2) , -∞ < Xi < ∞ .$
The relationship
 $P (Xi≤xi) = 12 erfc( - (xi-μi) 2 σi )$
is used, where erfc is the complementary error function, and is computed using s15adf.
When the two tail confidence probability is required the relationship
 $P (Xi≤|xi|) - P (Xi≤-|xi|) = erf( |xi-μi| 2 σi ) ,$
is used, where erf is the error function, and is computed using s15aef.
The input arrays to this routine are designed to allow maximum flexibility in the supply of vector arguments by re-using elements of any arrays that are shorter than the total number of evaluations required. See Section 2.6 in the G01 Chapter Introduction for further information.
NIST Digital Library of Mathematical Functions
Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth

## 5Arguments

1: $\mathbf{ltail}$Integer Input
On entry: the length of the array tail.
Constraint: ${\mathbf{ltail}}>0$.
2: $\mathbf{tail}\left({\mathbf{ltail}}\right)$Character(1) array Input
On entry: indicates which tail the returned probabilities should represent. Letting $Z$ denote a variate from a standard Normal distribution, and ${z}_{i}=\frac{{x}_{i}-{\mu }_{i}}{{\sigma }_{i}}$, then for , for $\mathit{i}=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{lx}},{\mathbf{ltail}},{\mathbf{lxmu}},{\mathbf{lxstd}}\right)$:
${\mathbf{tail}}\left(j\right)=\text{'L'}$
The lower tail probability is returned, i.e., ${p}_{i}=P\left(Z\le {z}_{i}\right)$.
${\mathbf{tail}}\left(j\right)=\text{'U'}$
The upper tail probability is returned, i.e., ${p}_{i}=P\left(Z\ge {z}_{i}\right)$.
${\mathbf{tail}}\left(j\right)=\text{'C'}$
The two tail (confidence interval) probability is returned, i.e., ${p}_{i}=P\left(Z\le |{z}_{i}|\right)-P\left(Z\le -|{z}_{i}|\right)$.
${\mathbf{tail}}\left(j\right)=\text{'S'}$
The two tail (significance level) probability is returned, i.e., ${p}_{i}=P\left(Z\ge |{z}_{i}|\right)+P\left(Z\le -|{z}_{i}|\right)$.
Constraint: ${\mathbf{tail}}\left(\mathit{j}\right)=\text{'L'}$, $\text{'U'}$, $\text{'C'}$ or $\text{'S'}$, for $\mathit{j}=1,2,\dots ,{\mathbf{ltail}}$.
3: $\mathbf{lx}$Integer Input
On entry: the length of the array x.
Constraint: ${\mathbf{lx}}>0$.
4: $\mathbf{x}\left({\mathbf{lx}}\right)$Real (Kind=nag_wp) array Input
On entry: ${x}_{i}$, the Normal variate values with ${x}_{i}={\mathbf{x}}\left(j\right)$, .
5: $\mathbf{lxmu}$Integer Input
On entry: the length of the array xmu.
Constraint: ${\mathbf{lxmu}}>0$.
6: $\mathbf{xmu}\left({\mathbf{lxmu}}\right)$Real (Kind=nag_wp) array Input
On entry: ${\mu }_{i}$, the means with ${\mu }_{i}={\mathbf{xmu}}\left(j\right)$, .
7: $\mathbf{lxstd}$Integer Input
On entry: the length of the array xstd.
Constraint: ${\mathbf{lxstd}}>0$.
8: $\mathbf{xstd}\left({\mathbf{lxstd}}\right)$Real (Kind=nag_wp) array Input
On entry: ${\sigma }_{i}$, the standard deviations with ${\sigma }_{i}={\mathbf{xstd}}\left(j\right)$, .
Constraint: ${\mathbf{xstd}}\left(\mathit{j}\right)>0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{lxstd}}$.
9: $\mathbf{p}\left(*\right)$Real (Kind=nag_wp) array Output
Note: the dimension of the array p must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{lx}},{\mathbf{ltail}},{\mathbf{lxmu}},{\mathbf{lxstd}}\right)$.
On exit: ${p}_{i}$, the probabilities for the Normal distribution.
10: $\mathbf{ivalid}\left(*\right)$Integer array Output
Note: the dimension of the array ivalid must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{lx}},{\mathbf{ltail}},{\mathbf{lxmu}},{\mathbf{lxstd}}\right)$.
On exit: ${\mathbf{ivalid}}\left(i\right)$ indicates any errors with the input arguments, with
${\mathbf{ivalid}}\left(i\right)=0$
No error.
${\mathbf{ivalid}}\left(i\right)=1$
On entry, invalid value supplied in tail when calculating ${p}_{i}$.
${\mathbf{ivalid}}\left(i\right)=2$
On entry, ${\sigma }_{i}\le 0.0$.
11: $\mathbf{ifail}$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:
${\mathbf{ifail}}=1$
On entry, at least one value of tail or xstd was invalid.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{ltail}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ltail}}>0$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{lx}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{lx}}>0$.
${\mathbf{ifail}}=4$
On entry, ${\mathbf{lxmu}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{lxmu}}>0$.
${\mathbf{ifail}}=5$
On entry, ${\mathbf{lxstd}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{lxstd}}>0$.
${\mathbf{ifail}}=-99$
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

g01saf is not threaded in any implementation.

None.

## 10Example

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

### 10.1Program Text

Program Text (g01safe.f90)

### 10.2Program Data

Program Data (g01safe.d)

### 10.3Program Results

Program Results (g01safe.r)