NAG Library Manual, Mark 27.2

Interfaces: FL CL CPP AD

NAG CL Interface Introduction

G01 (Stat) Chapter Contents

G01 (Stat) Chapter Introduction

g01ea: FL CL CPP AD

NAG CL Interface
g01eac (prob_normal)

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NAG Library Manual, Mark 27.2

Interfaces: FL CL CPP AD

NAG CL Interface Introduction

G01 (Stat) Chapter Contents

G01 (Stat) Chapter Introduction

g01ea: FL CL CPP AD

▸▿ Contents

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

© The Numerical Algorithms Group Ltd. 2021

CL Name Style:
Short (a00aac)
Long (impl_details)
Full (nag_info_impl_details)

1 Purpose

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

2 Specification

#include <nag.h>

double

g01eac (Nag_TailProbability tail, double x, NagError *fail)

The function may be called by the names: g01eac, nag_stat_prob_normal or nag_prob_normal.

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 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 s15aec.

4 References

NIST Digital Library of Mathematical Functions

Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth

5 Arguments

1: $tail$ – Nag_TailProbability Input

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

or

Nag_TwoTailConfid

.

2: $x$ – double Input

On entry:

x

, the value of the standard Normal variate.

3: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

If

fail . code =

NE_NOERROR, then g01eac returns

0.0

.

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

7 Accuracy

Accuracy is limited by machine precision. For detailed error analysis see s15adc and s15aec.

8 Parallelism and Performance

g01eac is not threaded in any implementation.

9 Further Comments

None.

10 Example

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

10.1 Program Text

Program Text (g01eace.c)

10.2 Program Data

Program Data (g01eace.d)

10.3 Program Results

Program Results (g01eace.r)

NAG Library Manual, Mark 27.2

Interfaces: FL CL CPP AD

NAG CL Interface Introduction

G01 (Stat) Chapter Contents

G01 (Stat) Chapter Introduction

g01ea: FL CL CPP AD

© The Numerical Algorithms Group Ltd. 2021