# NAG Library Function Document

## 1Purpose

nag_prob_chi_sq (g01ecc) returns the lower or upper tail probability for the ${\chi }^{2}$-distribution with real degrees of freedom.

## 2Specification

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

## 3Description

The lower tail probability for the ${\chi }^{2}$-distribution with $\nu$ degrees of freedom, $P\left(X\le x:\nu \right)$ is defined by:
 $PX≤x:ν=12ν/2Γν/2 ∫0.0xXν/2-1e-X/2dX, x≥0,ν>0.$
To calculate $P\left(X\le x:\nu \right)$ a transformation of a gamma distribution is employed, i.e., a ${\chi }^{2}$-distribution with $\nu$ degrees of freedom is equal to a gamma distribution with scale parameter $2$ and shape parameter $\nu /2$.

## 4References

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

## 5Arguments

1:    $\mathbf{tail}$Nag_TailProbabilityInput
On entry: indicates whether the upper or lower tail probability is required.
${\mathbf{tail}}=\mathrm{Nag_LowerTail}$
The lower tail probability is returned, i.e., $P\left(X\le x:\nu \right)$.
${\mathbf{tail}}=\mathrm{Nag_UpperTail}$
The upper tail probability is returned, i.e., $P\left(X\ge x:\nu \right)$.
Constraint: ${\mathbf{tail}}=\mathrm{Nag_LowerTail}$ or $\mathrm{Nag_UpperTail}$.
2:    $\mathbf{x}$doubleInput
On entry: $x$, the value of the ${\chi }^{2}$ variate with $\nu$ degrees of freedom.
Constraint: ${\mathbf{x}}\ge 0.0$.
3:    $\mathbf{df}$doubleInput
On entry: $\nu$, the degrees of freedom of the ${\chi }^{2}$-distribution.
Constraint: ${\mathbf{df}}>0.0$.
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_ALG_NOT_CONV
The series used to calculate the gamma probabilities has failed to converge. The result returned should represent an approximation to the solution.
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.
NE_REAL_ARG_LE
On entry, ${\mathbf{df}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{df}}>0.0$.
NE_REAL_ARG_LT
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{x}}\ge 0.0$.

## 7Accuracy

A relative accuracy of five significant figures is obtained in most cases.

## 8Parallelism and Performance

nag_prob_chi_sq (g01ecc) is not threaded in any implementation.

For higher accuracy the transformation described in Section 3 may be used with a direct call to nag_incomplete_gamma (s14bac).

## 10Example

Values from various ${\chi }^{2}$-distributions are read, the lower tail probabilities calculated, and all these values printed out, until the end of data is reached.

### 10.1Program Text

Program Text (g01ecce.c)

### 10.2Program Data

Program Data (g01ecce.d)

### 10.3Program Results

Program Results (g01ecce.r)

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