g01 Chapter Contents
g01 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_prob_normal_vector (g01sac)

## 1  Purpose

nag_prob_normal_vector (g01sac) returns a number of one or two tail probabilities for the Normal distribution.

## 2  Specification

 #include #include
 void nag_prob_normal_vector (Integer ltail, const Nag_TailProbability tail[], Integer lx, const double x[], Integer lxmu, const double xmu[], Integer lxstd, const double xstd[], double p[], Integer ivalid[], NagError *fail)

## 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 nag_erfc (s15adc).
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 nag_erf (s15aec).
The input arrays to this function 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.

## 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  Arguments

1:    $\mathbf{ltail}$IntegerInput
On entry: the length of the array tail.
Constraint: ${\mathbf{ltail}}>0$.
2:    $\mathbf{tail}\left[{\mathbf{ltail}}\right]$const Nag_TailProbabilityInput
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 $\mathit{i}=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{lx}},{\mathbf{ltail}},{\mathbf{lxmu}},{\mathbf{lxstd}}\right)$:
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_LowerTail}$
The lower tail probability is returned, i.e., ${p}_{i}=P\left(Z\le {z}_{i}\right)$.
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_UpperTail}$
The upper tail probability is returned, i.e., ${p}_{i}=P\left(Z\ge {z}_{i}\right)$.
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_TwoTailConfid}$
The two tail (confidence interval) probability is returned, i.e., ${p}_{i}=P\left(Z\le \left|{z}_{i}\right|\right)-P\left(Z\le -\left|{z}_{i}\right|\right)$.
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_TwoTailSignif}$
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: ${\mathbf{tail}}\left[\mathit{j}-1\right]=\mathrm{Nag_LowerTail}$, $\mathrm{Nag_UpperTail}$, $\mathrm{Nag_TwoTailConfid}$ or $\mathrm{Nag_TwoTailSignif}$, for $\mathit{j}=1,2,\dots ,{\mathbf{ltail}}$.
3:    $\mathbf{lx}$IntegerInput
On entry: the length of the array x.
Constraint: ${\mathbf{lx}}>0$.
4:    $\mathbf{x}\left[{\mathbf{lx}}\right]$const doubleInput
On entry: ${x}_{i}$, the Normal variate values with ${x}_{i}={\mathbf{x}}\left[j\right]$, .
5:    $\mathbf{lxmu}$IntegerInput
On entry: the length of the array xmu.
Constraint: ${\mathbf{lxmu}}>0$.
6:    $\mathbf{xmu}\left[{\mathbf{lxmu}}\right]$const doubleInput
On entry: ${\mu }_{i}$, the means with ${\mu }_{i}={\mathbf{xmu}}\left[j\right]$, .
7:    $\mathbf{lxstd}$IntegerInput
On entry: the length of the array xstd.
Constraint: ${\mathbf{lxstd}}>0$.
8:    $\mathbf{xstd}\left[{\mathbf{lxstd}}\right]$const doubleInput
On entry: ${\sigma }_{i}$, the standard deviations with ${\sigma }_{i}={\mathbf{xstd}}\left[j\right]$, .
Constraint: ${\mathbf{xstd}}\left[\mathit{j}-1\right]>0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{lxstd}}$.
9:    $\mathbf{p}\left[\mathit{dim}\right]$doubleOutput
Note: the dimension, dim, 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:  $\mathbf{ivalid}\left[\mathit{dim}\right]$IntegerOutput
Note: the dimension, dim, 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: ${\mathbf{ivalid}}\left[i-1\right]$ indicates any errors with the input arguments, with
${\mathbf{ivalid}}\left[i-1\right]=0$
No error.
${\mathbf{ivalid}}\left[i-1\right]=1$
 On entry, invalid value supplied in tail when calculating ${p}_{i}$.
${\mathbf{ivalid}}\left[i-1\right]=2$
 On entry, ${\sigma }_{i}\le 0.0$.
11:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_ARRAY_SIZE
On entry, ${\mathbf{ltail}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ltail}}>0$.
On entry, ${\mathbf{lx}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lx}}>0$.
On entry, ${\mathbf{lxmu}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lxmu}}>0$.
On entry, ${\mathbf{lxstd}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lxstd}}>0$.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
An unexpected error has been triggered by this function. Please contact NAG.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NW_IVALID
On entry, at least one value of tail or xstd was invalid.

## 7  Accuracy

Accuracy is limited by machine precision. For detailed error analysis see nag_erfc (s15adc) and nag_erf (s15aec).

## 8  Parallelism and Performance

nag_prob_normal_vector (g01sac) is not threaded in any implementation.

None.

## 10  Example

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

### 10.1  Program Text

Program Text (g01sace.c)

### 10.2  Program Data

Program Data (g01sace.d)

### 10.3  Program Results

Program Results (g01sace.r)