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_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.

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)