g01fa returns the deviate associated with the given probability of the standard Normal distribution.

Syntax

C#
public static double g01fa( string tail, double p, out int ifail )

Visual Basic
Public Shared Function g01fa ( _ tail As String, _ p As Double, _ <OutAttribute> ByRef ifail As Integer _ ) As Double

Visual C++
public: static double g01fa( String^ tail, double p, [OutAttribute] int% ifail )

F#
static member g01fa : tail : string * p : float * ifail : int byref -> float

Parameters

tail

Type: System..::..String

On entry: indicates which tail the supplied probability represents.

$tail = "L"$: The lower probability, i.e., $P (X \leq x_{p})$ .
$tail = "U"$: The upper probability, i.e., $P (X \geq x_{p})$ .
$tail = "S"$: The two tail (significance level) probability, i.e., $P (X \geq |x_{p}|) + P (X \leq - |x_{p}|)$ .
$tail = "C"$: The two tail (confidence interval) probability, i.e., $P (X \leq |x_{p}|) - P (X \leq - |x_{p}|)$ .

Constraint:

tail = "L"

"U"

"S"

"C"

p: Type: System..::..Double
On entry: $p$ , the probability from the standard Normal distribution as defined by tail.

Constraint: $0.0 < p < 1.0$ .

ifail: Type: System..::..Int32%
On exit: $ifail = 0$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Return Value

g01fa returns the deviate associated with the given probability of the standard Normal distribution.

Description

The deviate,

x_{p}

associated with the lower tail probability,

p

, for the standard Normal distribution is defined as the solution to

P (X \leq x_{p}) = p = \int_{- \infty}^{x_{p}} Z (X) d X,

where

Z (X) = \frac{1}{\sqrt{2 π}} e^{- X^{2} / 2}, - \infty < X < \infty .

The method used is an extension of that of Wichura (1988).

p

is first replaced by

q = p - 0.5

(a)

|q| \leq 0.3

x_{p}

is computed by a rational Chebyshev approximation

x_{p} = s \frac{A (s^{2})}{B (s^{2})},

where

s = \sqrt{2 π} q

and

A

B

are polynomials of degree

7

(b)

0.3 < |q| \leq 0.42

x_{p}

is computed by a rational Chebyshev approximation

x_{p} = sign q (\frac{C (t)}{D (t)}),

where

t = |q| - 0.3

and

C

D

are polynomials of degree

5

(c)

|q| > 0.42

x_{p}

is computed as

x_{p} = sign q [(\frac{E (u)}{F (u)}) + u],

where

u = \sqrt{- 2 \times \log (\min (p, 1 - p))}

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

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

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

Error Indicators and Warnings

Errors or warnings detected by the method:

If on exit

ifail \neq 0

, then g01fa returns

0.0

$ifail = 1$

On entry,

tail \neq "L"

"U"

"S"

"C"

$ifail = 2$

On entry,	$p \leq 0.0$ ,
or	$p \geq 1.0$ .

$ifail = -9000$: An error occured, see message report.

Accuracy

The accuracy is mainly limited by the machine precision.

Parallelism and Performance

None.

Further Comments

None.

Example

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

Example program (C#): g01fae.cs

Example program data: g01fae.d

Example program results: g01fae.r

Syntax

Parameters

Return Value

Description

References

Error Indicators and Warnings

Accuracy

Parallelism and Performance

Further Comments

Example

See Also