# NAG CL Interfaceg01sac (prob_​normal_​vector)

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## 1Purpose

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

## 2Specification

 #include
 void g01sac (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)
The function may be called by the names: g01sac, nag_stat_prob_normal_vector or nag_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 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 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.

## 4References

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: $\mathbf{tail}\left[{\mathbf{ltail}}\right]$const Nag_TailProbability 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 $\mathit{i}=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{lx}},{\mathbf{ltail}},{\mathbf{lxmu}},{\mathbf{lxstd}}\right)$:
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_LowerTail}$
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., ${p}_{i}=P\left(Z\le |{z}_{i}|\right)-P\left(Z\le -|{z}_{i}|\right)$.
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_TwoTailSignif}$
The two tail (significance level) probability is returned, i.e., ${p}_{i}=P\left(Z\ge |{z}_{i}|\right)+P\left(Z\le -|{z}_{i}|\right)$.
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: $\mathbf{lx}$Integer Input
On entry: the length of the array x.
Constraint: ${\mathbf{lx}}>0$.
4: $\mathbf{x}\left[{\mathbf{lx}}\right]$const double 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: $\mathbf{xmu}\left[{\mathbf{lxmu}}\right]$const double 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: $\mathbf{xstd}\left[{\mathbf{lxstd}}\right]$const double Input
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: $\mathbf{p}\left[\mathit{dim}\right]$double Output
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: $\mathbf{ivalid}\left[\mathit{dim}\right]$Integer Output
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 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NW_IVALID
On entry, at least one value of tail or xstd was invalid.

## 7Accuracy

Accuracy is limited by machine precision. For detailed error analysis see s15adc and s15aec.

## 8Parallelism and Performance

g01sac 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 (g01sace.c)

### 10.2Program Data

Program Data (g01sace.d)

### 10.3Program Results

Program Results (g01sace.r)