g01fc returns the deviate associated with the given lower tail probability of the ${\chi }^{2}$-distribution with real degrees of freedom.

# Syntax

C#
```public static double g01fc(
double p,
double df,
out int ifail
)```
Visual Basic
```Public Shared Function g01fc ( _
p As Double, _
df As Double, _
<OutAttribute> ByRef ifail As Integer _
) As Double```
Visual C++
```public:
static double g01fc(
double p,
double df,
[OutAttribute] int% ifail
)```
F#
```static member g01fc :
p : float *
df : float *
ifail : int byref -> float
```

#### Parameters

p
Type: System..::..Double
On entry: $p$, the lower tail probability from the required ${\chi }^{2}$-distribution.
Constraint: $0.0\le {\mathbf{p}}<1.0$.
df
Type: System..::..Double
On entry: $\nu$, the degrees of freedom of the ${\chi }^{2}$-distribution.
Constraint: ${\mathbf{df}}>0.0$.
ifail
Type: System..::..Int32%
On exit: ${\mathbf{ifail}}={0}$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

#### Return Value

g01fc returns the deviate associated with the given lower tail probability of the ${\chi }^{2}$-distribution with real degrees of freedom.

# Description

The deviate, ${x}_{p}$, associated with the lower tail probability $p$ of the ${\chi }^{2}$-distribution with $\nu$ degrees of freedom is defined as the solution to
 $PX≤xp:ν=p=12ν/2Γν/2∫0xpe-X/2Xv/2-1dX, 0≤xp<∞;ν>0.$
The required ${x}_{p}$ is found by using the relationship between a ${\chi }^{2}$-distribution and a gamma distribution, i.e., a ${\chi }^{2}$-distribution with $\nu$ degrees of freedom is equal to a gamma distribution with scale parameter $2$ and shape parameter $\nu /2$.
For very large values of $\nu$, greater than ${10}^{5}$, Wilson and Hilferty's normal approximation to the ${\chi }^{2}$ is used; see Kendall and Stuart (1969).

# References

Best D J and Roberts D E (1975) Algorithm AS 91. The percentage points of the ${\chi }^{2}$ distribution Appl. Statist. 24 385–388
Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth
Kendall M G and Stuart A (1969) The Advanced Theory of Statistics (Volume 1) (3rd Edition) Griffin

# Error Indicators and Warnings

Note: g01fc may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the method:
If ${\mathbf{ifail}}={1}$${2}$${3}$ or ${5}$ on exit, then g01fc returns $0.0$.
${\mathbf{ifail}}=1$
 On entry, ${\mathbf{p}}<0.0$, or ${\mathbf{p}}\ge 1.0$.
${\mathbf{ifail}}=2$
 On entry, ${\mathbf{df}}\le 0.0$.
${\mathbf{ifail}}=3$
p is too close to $0$ or $1$ for the result to be calculated.
${\mathbf{ifail}}=4$
The solution has failed to converge. The result should be a reasonable approximation.
${\mathbf{ifail}}=5$
The series used to calculate the gamma function has failed to converge. This is an unlikely error exit.
${\mathbf{ifail}}=-9000$
An error occured, see message report.

# Accuracy

The results should be accurate to five significant digits for most parameter values. Some accuracy is lost for $p$ close to $0.0$.

# Parallelism and Performance

None.

For higher accuracy the relationship described in [Description] may be used and a direct call to g01ff made.

# Example

This example reads lower tail probabilities for several ${\chi }^{2}$-distributions, and calculates and prints the corresponding deviates until the end of data is reached.

Example program (C#): g01fce.cs

Example program data: g01fce.d

Example program results: g01fce.r