\Header{Chisquare}{The (non-central) Chi-Square Distribution}
\alias{dchisq}{Chisquare}
\alias{pchisq}{Chisquare}
\alias{qchisq}{Chisquare}
\alias{rchisq}{Chisquare}
\keyword{distribution}{Chisquare}
\begin{Usage}
\begin{verbatim}
dchisq(x, df, ncp=0)
pchisq(q, df, ncp=0)
qchisq(p, df, ncp=0)
rchisq(n, df)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\texttt{x,q}] vector of quantiles.
\item[\texttt{p}] vector of probabilities.
\item[\texttt{n}] number of observations to generate.
\item[\texttt{df}] degrees of freedom.
\item[\texttt{ncp}] non-centrality parameter.
\end{ldescription}
\end{Arguments}
\begin{Value}
These functions provide information about the chi-square
(\eqn{\chi^2}{chi^2}) distribution with \texttt{df} degrees of freedom and
optional non-centrality parameter \texttt{ncp}.

The chi-square distribution with \texttt{df}\eqn{= n}{} degrees of freedom
has density
\deqn{f_n(x) = \frac{1}{{2}^{n/2} \Gamma (n/2)} {x}^{n/2-1} {e}^{-x/2}}{f_n(x) = 1 / (2^(n/2) Gamma(n/2))  x^(n/2-1) e^(-x/2)}
for \eqn{x > 0}{}.

\texttt{dchisq} gives the density \eqn{f_n}{},
\texttt{pchisq} gives the distribution function \eqn{F_n}{}, \texttt{qchisq} gives
the quantile function and \texttt{rchisq} generates random deviates.

The non-central chi-square distribution with \texttt{df}\eqn{= n}{} degrees of
freedom and non-centrality parameter \texttt{ncp} \eqn{= \lambda}{} has density
\deqn{
f(x) = e^{-\lambda / 2}
\sum_{r=0}^\infty \frac{(\lambda/2)^r}{r!}\, f_{n + 2r}(x)}{f(x) = exp(-lambda/2) SUM_{r=0}^infty ((lambda/2)^r / r!) dchisq(x, df + 2r)
}
for \eqn{x \ge 0}{}.\end{Value}
\begin{SeeAlso}
\texttt{\Link{dgamma}} for the gamma distribution which generalizes
the chi-square one.\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
dchisq(1, df=1:3)
pchisq(1, df= 3)
pchisq(1, df= 3, ncp = 0:4)# includes the above

x <- 1:10
## Chisquare( df = 2) is a special exponential distribution
dchisq(x, df=2) == dexp(x, 1/2)
pchisq(x, df=2) == pexp(x, 1/2)#- only approximately
\end{ExampleCode}
\end{Examples}

