g01 Chapter Contents
g01 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_deviates_normal_dist (g01cec)

## 1  Purpose

nag_deviates_normal_dist (g01cec) returns the deviate, ${x}_{p}$, associated with the given lower tail probability, $p$, of the standardized Normal distribution.

## 2  Specification

 #include #include
 double nag_deviates_normal_dist (double p, NagError *fail)

## 3  Description

${x}_{p}$ is calculated for the given $p$ such that
 $p = 1 2π ∫ -∞ x p e - u 2 / 2 du -∞ < x p < ∞ .$
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
 $x p = s A s 2 B s 2$
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
 $x p = sign⁡q C t D t$
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
 $x p = sign⁡q E u F u + u$
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.

## 4  References

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

## 5  Arguments

1:     pdoubleInput
On entry: the probability, $p$, from the standardized Normal distribution.
Constraint: $0.0<{\mathbf{p}}<1.0$.
2:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

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.
NE_REAL_ARG_GE
On entry, p must not be greater than or equal to $1.0$: ${\mathbf{p}}=〈\mathit{\text{value}}〉$.
NE_REAL_ARG_LE
On entry, p must not be less than or equal to $0.0$: ${\mathbf{p}}=〈\mathit{\text{value}}〉$.

## 7  Accuracy

nag_deviates_normal_dist (g01cec) attempts to attain a relative precision of $5.0×{10}^{-13}$.

If $X$ is a Normal random variable with mean $\mu$ and variance ${\sigma }^{2}$, the deviate corresponding to a lower tail probability of $p$ is $\mu +\sigma {x}_{p}$, where ${x}_{p}$ is the standardized Normal deviate returned by nag_deviates_normal_dist (g01cec).

## 9  Example

The deviates corresponding to several lower tail probabilities from the standard Normal distribution are calculated and printed.

### 9.1  Program Text

Program Text (g01cece.c)

None.

### 9.3  Program Results

Program Results (g01cece.r)