NAG Library Function Document

nag_prob_normal (g01eac)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

nag_prob_normal (g01eac) returns a one or two tail probability for the standard Normal distribution.

2 Specification

#include <nag.h>

#include <nagg01.h>

double

nag_prob_normal (Nag_TailProbability tail, double x, NagError *fail)

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 nag_erfc (s15adc). 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 nag_erf (s15aec).

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 Arguments

1: $tail$ – Nag_TailProbabilityInput

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

$tail = Nag_LowerTail$: The lower tail probability is returned, i.e., $P (X \leq x)$ .
$tail = Nag_UpperTail$: The upper tail probability is returned, i.e., $P (X \geq x)$ .
$tail = Nag_TwoTailSignif$: The two tail (significance level) probability is returned, i.e., $P (X \geq |x|) + P (X \leq - |x|)$ .
$tail = Nag_TwoTailConfid$: The two tail (confidence interval) probability is returned, i.e., $P (X \leq |x|) - P (X \leq - |x|)$ .

Constraint:

tail = Nag_LowerTail

Nag_UpperTail

Nag_TwoTailSignif

Nag_TwoTailConfid

2: $x$ – doubleInput

On entry:

x

, the value of the standard Normal variate.

3: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ 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.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.

7 Accuracy

Accuracy is limited by machine precision. For detailed error analysis see nag_erfc (s15adc) and nag_erf (s15aec).

8 Parallelism and Performance

Not applicable.

9 Further Comments

None.

10 Example

Four values of tail and x are input and the probabilities calculated and printed.

NAG Library Function Documentnag_prob_normal (g01eac)