# Chisq-class

##### Class "Chisq"

The chi-squared distribution with `df`

\(= n\) degrees of
freedom has density
$$f_n(x) = \frac{1}{{2}^{n/2} \Gamma (n/2)} {x}^{n/2-1} {e}^{-x/2}$$
for \(x > 0\). The mean and variance are \(n\) and \(2n\).

The non-central chi-squared distribution with `df`

\(= n\)
degrees of freedom and non-centrality parameter `ncp`

\(= \lambda\) has density
$$
f(x) = e^{-\lambda / 2}
\sum_{r=0}^\infty \frac{(\lambda/2)^r}{r!}\, f_{n + 2r}(x)$$
for \(x \ge 0\). For integer \(n\), this is the distribution of
the sum of squares of \(n\) normals each with variance one,
\(\lambda\) being the sum of squares of the normal means.

C.f. `rchisq`

- Keywords
- distribution

##### Note

Warning: The code for pchisq and qchisq is unreliable for values of ncp above approximately 290.

##### Objects from the Class

Objects can be created by calls of the form `Chisq(df, ncp)`

.
This object is a chi-squared distribution.

##### Slots

`img`

Object of class

`"Reals"`

: The space of the image of this distribution has got dimension 1 and the name "Real Space".`param`

Object of class

`"ChisqParameter"`

: the parameter of this distribution (df and ncp), declared at its instantiation`r`

Object of class

`"function"`

: generates random numbers (calls function rchisq)`d`

Object of class

`"function"`

: density function (calls function dchisq)`p`

Object of class

`"function"`

: cumulative function (calls function pchisq)`q`

Object of class

`"function"`

: inverse of the cumulative function (calls function qchisq)`.withArith`

logical: used internally to issue warnings as to interpretation of arithmetics

`.withSim`

logical: used internally to issue warnings as to accuracy

`.logExact`

logical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function

`.lowerExact`

logical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function

`Symmetry`

object of class

`"DistributionSymmetry"`

; used internally to avoid unnecessary calculations.

##### Extends

Class `"ExpOrGammaOrChisq"`

, directly.
Class `"AbscontDistribution"`

, by class `"ExpOrGammaOrChisq"`

.
Class `"UnivariateDistribution"`

, by class `"AbscontDistribution"`

.
Class `"Distribution"`

, by class `"UnivariateDistribution"`

.

##### Is-Relations

By means of `setIs`

, R ``knows'' that a distribution object `obj`

of class `"Chisq"`

with non-centrality 0 also is
a Gamma distribution with parameters `shape = df(obj)/2, scale = 2`

.

##### Methods

- initialize
`signature(.Object = "Chisq")`

: initialize method- df
`signature(object = "Chisq")`

: returns the slot df of the parameter of the distribution- df<-
`signature(object = "Chisq")`

: modifies the slot df of the parameter of the distribution- ncp
`signature(object = "Chisq")`

: returns the slot ncp of the parameter of the distribution- ncp<-
`signature(object = "Chisq")`

: modifies the slot ncp of the parameter of the distribution- +
`signature(e1 = "Chisq", e2 = "Chisq")`

: For the chi-squared distribution we use its closedness under convolutions.

##### See Also

`ChisqParameter-class`

`AbscontDistribution-class`

`Reals-class`

`rchisq`

##### Examples

```
# NOT RUN {
C <- Chisq(df = 1, ncp = 1) # C is a chi-squared distribution with df=1 and ncp=1.
r(C)(1) # one random number generated from this distribution, e.g. 0.2557184
d(C)(1) # Density of this distribution is 0.2264666 for x = 1.
p(C)(1) # Probability that x < 1 is 0.4772499.
q(C)(.1) # Probability that x < 0.04270125 is 0.1.
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)
df(C) # df of this distribution is 1.
df(C) <- 2 # df of this distribution is now 2.
is(C, "Gammad") # no
C0 <- Chisq() # default: Chisq(df=1,ncp=0)
is(C0, "Gammad") # yes
as(C0,"Gammad")
# }
```

*Documentation reproduced from package distr, version 2.8.0, License: LGPL-3*