\section{\module{random} ---
         Generate pseudo-random numbers}

\declaremodule{standard}{random}
\modulesynopsis{Generate pseudo-random numbers with various common
                distributions.}


This module implements pseudo-random number generators for various
distributions.
For integers, uniform selection from a range.
For sequences, uniform selection of a random element, and a function to
generate a random permutation of a list in-place.
On the real line, there are functions to compute uniform, normal (Gaussian),
lognormal, negative exponential, gamma, and beta distributions.
For generating distribution of angles, the circular uniform and
von Mises distributions are available.

Almost all module functions depend on the basic function
\function{random()}, which generates a random float uniformly in
the semi-open range [0.0, 1.0).  Python uses the standard Wichmann-Hill
generator, combining three pure multiplicative congruential
generators of modulus 30269, 30307 and 30323.  Its period (how many
numbers it generates before repeating the sequence exactly) is
6,953,607,871,644.  While of much higher quality than the \function{rand()}
function supplied by most C libraries, the theoretical properties
are much the same as for a single linear congruential generator of
large modulus.

The functions in this module are not threadsafe:  if you want to call these
functions from multiple threads, you should explicitly serialize the calls.
Else, because no critical sections are implemented internally, calls
from different threads may see the same return values.


\begin{funcdesc}{seed}{\optional{x}}
  Initialize the basic random number generator.
  Optional argument \var{x} can be any hashable object,
  and the generator is seeded from its hash code.
  It is not guaranteed that distinct hash codes will produce distinct
  seeds.
  If \var{x} is omitted or \code{None},
  the seed is derived from the current system time.
  The seed is also set from the current system time when
  the module is first imported.
\end{funcdesc}

\begin{funcdesc}{choice}{seq}
  Return a random element from the non-empty sequence \var{seq}.
\end{funcdesc}

\begin{funcdesc}{randint}{a, b}
  \deprecated{2.0}{Use \function{randrange()} instead.}
  Return a random integer \var{N} such that
  \code{\var{a} <= \var{N} <= \var{b}}.
\end{funcdesc}

\begin{funcdesc}{randrange}{\optional{start,} stop\optional{, step}}
  Return a randomly selected element from \code{range(\var{start},
  \var{stop}, \var{step})}.  This is equivalent to
  \code{choice(range(\var{start}, \var{stop}, \var{step}))},
  but doesn't actually build a range object.
  \versionadded{1.5.2}
\end{funcdesc}

\begin{funcdesc}{random}{}
  Return the next random floating point number in the range [0.0, 1.0).
\end{funcdesc}

\begin{funcdesc}{uniform}{a, b}
  Return a random real number \var{N} such that
  \code{\var{a} <= \var{N} < \var{b}}.
\end{funcdesc}


The following functions are defined to support specific distributions,
and all return real values.  Function parameters are named after the
corresponding variables in the distribution's equation, as used in
common mathematical practice; most of these equations can be found in
any statistics text.


\begin{funcdesc}{betavariate}{alpha, beta}
  Beta distribution.  Conditions on the parameters are
  \code{\var{alpha} > -1} and \code{\var{beta} > -1}.
  Returned values range between 0 and 1.
\end{funcdesc}

\begin{funcdesc}{cunifvariate}{mean, arc}
  Circular uniform distribution.  \var{mean} is the mean angle, and
  \var{arc} is the range of the distribution, centered around the mean
  angle.  Both values must be expressed in radians, and can range
  between 0 and \emph{pi}.  Returned values range between
  \code{\var{mean} - \var{arc}/2} and \code{\var{mean} +
  \var{arc}/2}.
\end{funcdesc}

\begin{funcdesc}{expovariate}{lambd}
  Exponential distribution.  \var{lambd} is 1.0 divided by the desired
  mean.  (The parameter would be called ``lambda'', but that is a
  reserved word in Python.)  Returned values range from 0 to
  positive infinity.
\end{funcdesc}

\begin{funcdesc}{gamma}{alpha, beta}
  Gamma distribution.  (\emph{Not} the gamma function!)  Conditions on
  the parameters are \code{\var{alpha} > -1} and \code{\var{beta} > 0}.
\end{funcdesc}

\begin{funcdesc}{gauss}{mu, sigma}
  Gaussian distribution.  \var{mu} is the mean, and \var{sigma} is the
  standard deviation.  This is slightly faster than the
  \function{normalvariate()} function defined below.
\end{funcdesc}

\begin{funcdesc}{lognormvariate}{mu, sigma}
  Log normal distribution.  If you take the natural logarithm of this
  distribution, you'll get a normal distribution with mean \var{mu}
  and standard deviation \var{sigma}.  \var{mu} can have any value,
  and \var{sigma} must be greater than zero.
\end{funcdesc}

\begin{funcdesc}{normalvariate}{mu, sigma}
  Normal distribution.  \var{mu} is the mean, and \var{sigma} is the
  standard deviation.
\end{funcdesc}

\begin{funcdesc}{vonmisesvariate}{mu, kappa}
  \var{mu} is the mean angle, expressed in radians between 0 and
  2*\emph{pi}, and \var{kappa} is the concentration parameter, which
  must be greater than or equal to zero.  If \var{kappa} is equal to
  zero, this distribution reduces to a uniform random angle over the
  range 0 to 2*\emph{pi}.
\end{funcdesc}

\begin{funcdesc}{paretovariate}{alpha}
  Pareto distribution.  \var{alpha} is the shape parameter.
\end{funcdesc}

\begin{funcdesc}{weibullvariate}{alpha, beta}
  Weibull distribution.  \var{alpha} is the scale parameter and
  \var{beta} is the shape parameter.
\end{funcdesc}


This function does not represent a specific distribution, but
implements a standard useful algorithm:

\begin{funcdesc}{shuffle}{x\optional{, random}}
  Shuffle the sequence \var{x} in place.
  The optional argument \var{random} is a 0-argument function
  returning a random float in [0.0, 1.0); by default, this is the
  function \function{random()}.

  Note that for even rather small \code{len(\var{x})}, the total
  number of permutations of \var{x} is larger than the period of most
  random number generators; this implies that most permutations of a
  long sequence can never be generated.
\end{funcdesc}

\begin{seealso}
  \seetext{Wichmann, B. A. \& Hill, I. D., ``Algorithm AS 183:
           An efficient and portable pseudo-random number generator'',
           \citetitle{Applied Statistics} 31 (1982) 188-190.}
\end{seealso}