\HeaderA{Logistic}{The Logistic Distribution}{Logistic}
\aliasA{dlogis}{Logistic}{dlogis}
\aliasA{plogis}{Logistic}{plogis}
\aliasA{qlogis}{Logistic}{qlogis}
\aliasA{rlogis}{Logistic}{rlogis}
\keyword{distribution}{Logistic}
\begin{Description}\relax
Density, distribution function, quantile function and random
generation for the logistic distribution with parameters
\code{location} and \code{scale}.
\end{Description}
\begin{Usage}
\begin{verbatim}
dlogis(x, location = 0, scale = 1, log = FALSE)
plogis(q, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)
qlogis(p, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)
rlogis(n, location = 0, scale = 1)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{x, q}] vector of quantiles.
\item[\code{p}] vector of probabilities.
\item[\code{n}] number of observations. If \code{length(n) > 1}, the length
is taken to be the number required.
\item[\code{location, scale}] location and scale parameters.
\item[\code{log, log.p}] logical; if TRUE, probabilities p are given as log(p).
\item[\code{lower.tail}] logical; if TRUE (default), probabilities are
\eqn{P[X \le x]}{P[X <= x]}, otherwise, \eqn{P[X > x]}{P[X > x]}.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
If \code{location} or \code{scale} are omitted, they assume the
default values of \code{0} and \code{1} respectively.

The Logistic distribution with \code{location} \eqn{= \mu}{= m} and
\code{scale} \eqn{= \sigma}{= s} has distribution function
\deqn{
F(x) = \frac{1}{1 + e^{-(x-\mu)/\sigma}}
}{F(x) = 1 / (1 + exp(-(x-m)/s))}  and density
\deqn{
f(x)= \frac{1}{\sigma}\frac{e^{(x-\mu)/\sigma}}{(1 + e^{(x-\mu)/\sigma})^2}
}{f(x) = 1/s exp((x-m)/s) (1 + exp((x-m)/s))^-2.}

It is a long-tailed distribution with mean \eqn{\mu}{m} and variance
\eqn{\pi^2/3 \sigma^2}{pi^2 /3 s^2}.
\end{Details}
\begin{Value}
\code{dlogis} gives the density,
\code{plogis} gives the distribution function,
\code{qlogis} gives the quantile function, and
\code{rlogis} generates random deviates.
\end{Value}
\begin{Note}\relax
\code{qlogis(p)} is the same as the well known \sQuote{\emph{logit}}
function, \eqn{logit(p) = \log(p/(1-p))}{}, and \code{plogis(x)} has
consequently been called the \sQuote{inverse logit}.

The distribution function is a rescaled hyperbolic tangent,
\code{plogis(x) == (1+ \LinkA{tanh}{tanh}(x/2))/2}, and it is called a
\emph{sigmoid function} in contexts such as neural networks.
\end{Note}
\begin{Source}\relax
\code{[dpr]logis} are calculated directly from the definitions.

\code{rlogis} uses inversion.
\end{Source}
\begin{References}\relax
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988)
\emph{The New S Language}.
Wadsworth \& Brooks/Cole.

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995)
\emph{Continuous Univariate Distributions}, volume 2, chapter 23.
Wiley, New York.
\end{References}
\begin{Examples}
\begin{ExampleCode}
var(rlogis(4000, 0, scale = 5))# approximately (+/- 3)
pi^2/3 * 5^2
\end{ExampleCode}
\end{Examples}

