# NAG C Library Function Document

## 1Purpose

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

## 2Specification

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

## 3Description

The lower tail probability for the standard Normal distribution, $P\left(X\le x\right)$ is defined by:
 $PX≤x=∫-∞xZXdX,$
where
 $ZX=12π e-X2/2, -∞
The relationship
 $PX≤x=12erfc-x2$
is used, where erfc is the complementary error function, and is computed using nag_erfc (s15adc). For the upper tail probability the relationship $P\left(X\ge x\right)=P\left(X\le -x\right)$ 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
 $PX≤x-PX≤-x=erfx2 ,$
is used, where erf is the error function, and is computed using nag_erf (s15aec).

## 4References

NIST Digital Library of Mathematical Functions
Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth

## 5Arguments

1:    $\mathbf{tail}$Nag_TailProbabilityInput
On entry: indicates which tail the returned probability should represent.
${\mathbf{tail}}=\mathrm{Nag_LowerTail}$
The lower tail probability is returned, i.e., $P\left(X\le x\right)$.
${\mathbf{tail}}=\mathrm{Nag_UpperTail}$
The upper tail probability is returned, i.e., $P\left(X\ge x\right)$.
${\mathbf{tail}}=\mathrm{Nag_TwoTailSignif}$
The two tail (significance level) probability is returned, i.e., $P\left(X\ge \left|x\right|\right)+P\left(X\le -\left|x\right|\right)$.
${\mathbf{tail}}=\mathrm{Nag_TwoTailConfid}$
The two tail (confidence interval) probability is returned, i.e., $P\left(X\le \left|x\right|\right)-P\left(X\le -\left|x\right|\right)$.
Constraint: ${\mathbf{tail}}=\mathrm{Nag_LowerTail}$, $\mathrm{Nag_UpperTail}$, $\mathrm{Nag_TwoTailSignif}$ or $\mathrm{Nag_TwoTailConfid}$.
2:    $\mathbf{x}$doubleInput
On entry: $x$, the value of the standard Normal variate.
3:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

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

## 8Parallelism and Performance

nag_prob_normal (g01eac) is not threaded in any implementation.

None.

## 10Example

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

### 10.1Program Text

Program Text (g01eace.c)

### 10.2Program Data

Program Data (g01eace.d)

### 10.3Program Results

Program Results (g01eace.r)