NAG Library Routine Document

G01EAF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

G01EAF returns a one or two tail probability for the standard Normal distribution, via the routine name.

2 Specification

FUNCTION G01EAF (

TAIL, X, IFAIL)

REAL (KIND=nag_wp) G01EAF

INTEGER	IFAIL
REAL (KIND=nag_wp)	X
CHARACTER(1)	TAIL

3 Description

The lower tail probability for the standard Normal distribution,

P (X \leq x)

is defined by:

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

where

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

The relationship

P (X \leq x) = \frac{1}{2} erfc (\frac{- x}{\sqrt{2}})

is used, where erfc is the complementary error function, and is computed using S15ADF. For the upper tail probability the relationship

P (X \geq x) = P (X \leq - x)

is used and for the two tail significance level probability twice the probability obtained from the absolute value of

x

is returned.

When the two tail confidence probability is required the relationship

P (X \leq |x|) - P (X \leq - |x|) = \erf (\frac{|x|}{\sqrt{2}}),

is used, where erf is the error function, and is computed using S15AEF.

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 Parameters

1: TAIL – CHARACTER(1)Input

On entry: indicates which tail the returned probability should represent.

$TAIL ='L'$: The lower tail probability is returned, i.e., $P (X \leq x)$ .
$TAIL ='U'$: The upper tail probability is returned, i.e., $P (X \geq x)$ .
$TAIL ='S'$: The two tail (significance level) probability is returned, i.e., $P (X \geq |x|) + P (X \leq - |x|)$ .
$TAIL ='C'$: The two tail (confidence interval) probability is returned, i.e., $P (X \leq |x|) - P (X \leq - |x|)$ .

Constraint:

TAIL ='L'

'U'

'S'

'C'

2: X – REAL (KIND=nag_wp)Input

On entry:

x

, the value of the standard Normal variate.

3: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

IFAIL \neq 0

, then G01EAF returns

0.0

$IFAIL = 1$

On entry,

TAIL \neq'L'

'U'

'S'

'C'

7 Accuracy

Accuracy is limited by machine precision. For detailed error analysis see S15ADF and S15AEF.

8 Further Comments

None.

9 Example

Four values of TAIL and X are input and the probabilities calculated and printed.

NAG Library Routine DocumentG01EAF