G01 Chapter Contents
G01 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentG01SAF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

G01SAF returns a number of one or two tail probabilities for the Normal distribution.

## 2  Specification

 SUBROUTINE G01SAF ( LTAIL, TAIL, LX, X, LXMU, XMU, LXSTD, XSTD, P, IVALID, IFAIL)
 INTEGER LTAIL, LX, LXMU, LXSTD, IVALID(*), IFAIL REAL (KIND=nag_wp) X(LX), XMU(LXMU), XSTD(LXSTD), P(*) CHARACTER(1) TAIL(LTAIL)

## 3  Description

The lower tail probability for the Normal distribution, $P\left({X}_{i}\le {x}_{i}\right)$ is defined by:
 $PXi≤xi = ∫ -∞ xi ZiXidXi ,$
where
 $ZiXi = 1 2πσi2 e -Xi-μi2/2σi2 , -∞ < Xi < ∞ .$
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 parameters 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.

## 4  References

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

## 5  Parameters

1:     LTAIL – INTEGERInput
On entry: the length of the array TAIL.
Constraint: ${\mathbf{LTAIL}}>0$.
2:     TAIL(LTAIL) – CHARACTER(1) arrayInput
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., ${p}_{i}=P\left(Z\le \left|{z}_{i}\right|\right)-P\left(Z\le -\left|{z}_{i}\right|\right)$.
${\mathbf{TAIL}}\left(j\right)=\text{'S'}$
The two tail (significance level) probability is returned, i.e., ${p}_{i}=P\left(Z\ge \left|{z}_{i}\right|\right)+P\left(Z\le -\left|{z}_{i}\right|\right)$.
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:     LX – INTEGERInput
On entry: the length of the array X.
Constraint: ${\mathbf{LX}}>0$.
4:     X(LX) – REAL (KIND=nag_wp) arrayInput
On entry: ${x}_{i}$, the Normal variate values with ${x}_{i}={\mathbf{X}}\left(j\right)$, .
5:     LXMU – INTEGERInput
On entry: the length of the array XMU.
Constraint: ${\mathbf{LXMU}}>0$.
6:     XMU(LXMU) – REAL (KIND=nag_wp) arrayInput
On entry: ${\mu }_{i}$, the means with ${\mu }_{i}={\mathbf{XMU}}\left(j\right)$, .
7:     LXSTD – INTEGERInput
On entry: the length of the array XSTD.
Constraint: ${\mathbf{LXSTD}}>0$.
8:     XSTD(LXSTD) – REAL (KIND=nag_wp) arrayInput
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:     P($*$) – REAL (KIND=nag_wp) arrayOutput
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:   IVALID($*$) – INTEGER arrayOutput
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:   IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction 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 parameter, 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).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, 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: ${\mathbf{LXSTD}}>0$.
${\mathbf{IFAIL}}=-999$
Dynamic memory allocation failed.

## 7  Accuracy

Accuracy is limited by machine precision. For detailed error analysis see S15ADF and S15AEF.

None.

## 9  Example

Four values of TAIL, X, XMU and XSTD are input and the probabilities calculated and printed.

### 9.1  Program Text

Program Text (g01safe.f90)

### 9.2  Program Data

Program Data (g01safe.d)

### 9.3  Program Results

Program Results (g01safe.r)