# NAG Library Function Document

## 1Purpose

nag_normal_pdf (g01kac) returns the value of the probability density function (PDF) for the Normal (Gaussian) distribution with mean $\mu$ and variance ${\sigma }^{2}$ at a point $x$.

## 2Specification

 #include #include
 double nag_normal_pdf (double x, double xmean, double xstd, NagError *fail)

## 3Description

The Normal distribution has probability density function (PDF)
 $fx = 1 σ ⁢ 2π e -x-μ2/2σ2 , σ>0 .$

None.

## 5Arguments

1:    $\mathbf{x}$doubleInput
On entry: $x$, the value at which the PDF is to be evaluated.
2:    $\mathbf{xmean}$doubleInput
On entry: $\mu$, the mean of the Normal distribution.
3:    $\mathbf{xstd}$doubleInput
On entry: $\sigma$, the standard deviation of the Normal distribution.
Constraint: $z<{\mathbf{xstd}}\sqrt{2\pi }<1.0/z$, where $z={\mathbf{nag_real_safe_small_number}}$, the safe range parameter.
4:    $\mathbf{fail}$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.
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.
NE_OVERFLOW
Computation abandoned owing to an internal calculation overflowing.
NE_REAL
On entry, ${\mathbf{xstd}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{xstd}}×\sqrt{2.0\pi }>U$, where $U$ is the safe range parameter as defined by nag_real_safe_small_number (X02AMC).
NE_UNDERFLOW
Computation abandoned owing to underflow of $\frac{1}{\left(\sigma ×\sqrt{2\pi }\right)}$.

Not applicable.

## 8Parallelism and Performance

nag_normal_pdf (g01kac) is not threaded in any implementation.

None.

## 10Example

This example prints the value of the Normal distribution PDF at four different points x with differing xmean and xstd.

### 10.1Program Text

Program Text (g01kace.c)

### 10.2Program Data

Program Data (g01kace.d)

### 10.3Program Results

Program Results (g01kace.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017