# NAG CL Interfaceg01ecc (prob_​chisq)

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## 1Purpose

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

## 2Specification

 #include
 double g01ecc (Nag_TailProbability tail, double x, double df, NagError *fail)
The function may be called by the names: g01ecc, nag_stat_prob_chisq or nag_prob_chi_sq.

## 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:
 $P(X≤x:ν)=12ν/2Γ(ν/2) ∫0.0xXν/2-1e-X/2dX, x≥0,ν>0.$
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

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

## 5Arguments

1: $\mathbf{tail}$Nag_TailProbability Input
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: $\mathbf{x}$double Input
On entry: $x$, the value of the ${\chi }^{2}$ variate with $\nu$ degrees of freedom.
Constraint: ${\mathbf{x}}\ge 0.0$.
3: $\mathbf{df}$double Input
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 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

If ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_CHARACTER, NE_REAL_ARG_LE or NE_REAL_ARG_LT on exit, then g01ecc returns $0.0$.
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 3.1.2 in the Introduction to the NAG Library CL Interface 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 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.
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

Background information to multithreading can be found in the Multithreading documentation.
g01ecc is not threaded in any implementation.

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