# NAG Library Function Document

## 1Purpose

nag_deviates_normal (g01fac) returns the deviate associated with the given probability of the standard Normal distribution.

## 2Specification

 #include #include
 double nag_deviates_normal (Nag_TailProbability tail, double p, NagError *fail)

## 3Description

The deviate, ${x}_{p}$ associated with the lower tail probability, $p$, for the standard Normal distribution is defined as the solution to
 $PX≤xp=p=∫-∞xpZXdX,$
where
 $ZX=12πe-X2/2, -∞
The method used is an extension of that of Wichura (1988). $p$ is first replaced by $q=p-0.5$.
(a) If $\left|q\right|\le 0.3$, ${x}_{p}$ is computed by a rational Chebyshev approximation
 $xp=sAs2 Bs2 ,$
where $s=\sqrt{2\pi }q$ and $A$, $B$ are polynomials of degree $7$.
(b) If $0.3<\left|q\right|\le 0.42$, ${x}_{p}$ is computed by a rational Chebyshev approximation
 $xp=sign⁡q Ct Dt ,$
where $t=\left|q\right|-0.3$ and $C$, $D$ are polynomials of degree $5$.
(c) If $\left|q\right|>0.42$, ${x}_{p}$ is computed as
 $xp=sign⁡q Eu Fu +u ,$
where $u=\sqrt{-2×\mathrm{log}\left(\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(p,1-p\right)\right)}$ and $E$, $F$ are polynomials of degree $6$.
For the upper tail probability $-{x}_{p}$ is returned, while for the two tail probabilities the value ${x}_{{p}^{*}}$ is returned, where ${p}^{*}$ is the required tail probability computed from the input value of $p$.

## 4References

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
Wichura (1988) Algorithm AS 241: the percentage points of the Normal distribution Appl. Statist. 37 477–484

## 5Arguments

1:    $\mathbf{tail}$Nag_TailProbabilityInput
On entry: indicates which tail the supplied probability represents.
${\mathbf{tail}}=\mathrm{Nag_LowerTail}$
The lower probability, i.e., $P\left(X\le {x}_{p}\right)$.
${\mathbf{tail}}=\mathrm{Nag_UpperTail}$
The upper probability, i.e., $P\left(X\ge {x}_{p}\right)$.
${\mathbf{tail}}=\mathrm{Nag_TwoTailSignif}$
The two tail (significance level) probability, i.e., $P\left(X\ge \left|{x}_{p}\right|\right)+P\left(X\le -\left|{x}_{p}\right|\right)$.
${\mathbf{tail}}=\mathrm{Nag_TwoTailConfid}$
The two tail (confidence interval) probability, i.e., $P\left(X\le \left|{x}_{p}\right|\right)-P\left(X\le -\left|{x}_{p}\right|\right)$.
Constraint: ${\mathbf{tail}}=\mathrm{Nag_LowerTail}$, $\mathrm{Nag_UpperTail}$, $\mathrm{Nag_TwoTailSignif}$ or $\mathrm{Nag_TwoTailConfid}$.
2:    $\mathbf{p}$doubleInput
On entry: $p$, the probability from the standard Normal distribution as defined by tail.
Constraint: $0.0<{\mathbf{p}}<1.0$.
3:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

If on exit ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, then nag_deviates_normal (g01fac) returns $0.0$.
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.
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 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.
NE_REAL_ARG_GE
On entry, ${\mathbf{p}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{p}}<1.0$.
NE_REAL_ARG_LE
On entry, ${\mathbf{p}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{p}}>0.0$.

## 7Accuracy

The accuracy is mainly limited by the machine precision.

## 8Parallelism and Performance

nag_deviates_normal (g01fac) is not threaded in any implementation.

None.

## 10Example

Four values of tail and p are input and the deviates calculated and printed.

### 10.1Program Text

Program Text (g01face.c)

### 10.2Program Data

Program Data (g01face.d)

### 10.3Program Results

Program Results (g01face.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017