\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, a function to
generate a random permutation of a list in-place, and a function for
random sampling without replacement.

On the real line, there are functions to compute uniform, normal (Gaussian),
lognormal, negative exponential, gamma, and beta distributions.
For generating distributions of angles, the von Mises distribution
is 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 Mersenne Twister as
the core generator.  It produces 53-bit precision floats and has a
period of 2**19937-1.  The underlying implementation in C
is both fast and threadsafe.  The Mersenne Twister is one of the most
extensively tested random number generators in existence.  However, being
completely deterministic, it is not suitable for all purposes, and is
completely unsuitable for cryptographic purposes.

The functions supplied by this module are actually bound methods of a
hidden instance of the \class{random.Random} class.  You can
instantiate your own instances of \class{Random} to get generators
that don't share state.  This is especially useful for multi-threaded
programs, creating a different instance of \class{Random} for each
thread, and using the \method{jumpahead()} method to make it likely that the
generated sequences seen by each thread don't overlap.

Class \class{Random} can also be subclassed if you want to use a
different basic generator of your own devising: in that case, override
the \method{random()}, \method{seed()}, \method{getstate()},
\method{setstate()} and \method{jumpahead()} methods.
Optionally, a new generator can supply a \method{getrandombits()}
method --- this allows \method{randrange()} to produce selections
over an arbitrarily large range.
\versionadded[the \method{getrandombits()} method]{2.4}

As an example of subclassing, the \module{random} module provides
the \class{WichmannHill} class that implements an alternative generator
in pure Python.  The class provides a backward compatible way to
reproduce results from earlier versions of Python, which used the
Wichmann-Hill algorithm as the core generator.  Note that this Wichmann-Hill
generator can no longer be recommended:  its period is too short by
contemporary standards, and the sequence generated is known to fail some
stringent randomness tests.  See the references below for a recent
variant that repairs these flaws.
\versionchanged[Substituted MersenneTwister for Wichmann-Hill]{2.3}


Bookkeeping functions:

\begin{funcdesc}{seed}{\optional{x}}
  Initialize the basic random number generator.
  Optional argument \var{x} can be any hashable object.
  If \var{x} is omitted or \code{None}, current system time is used;
  current system time is also used to initialize the generator when the
  module is first imported.  If randomness sources are provided by the
  operating system, they are used instead of the system time (see the
  \function{os.urandom()}
  function for details on availability).  \versionchanged[formerly,
  operating system resources were not used]{2.4}
  If \var{x} is not \code{None} or an int or long,
  \code{hash(\var{x})} is used instead.
  If \var{x} is an int or long, \var{x} is used directly.
\end{funcdesc}

\begin{funcdesc}{getstate}{}
  Return an object capturing the current internal state of the
  generator.  This object can be passed to \function{setstate()} to
  restore the state.
  \versionadded{2.1}
\end{funcdesc}

\begin{funcdesc}{setstate}{state}
  \var{state} should have been obtained from a previous call to
  \function{getstate()}, and \function{setstate()} restores the
  internal state of the generator to what it was at the time
  \function{setstate()} was called.
  \versionadded{2.1}
\end{funcdesc}

\begin{funcdesc}{jumpahead}{n}
  Change the internal state to one different from and likely far away from
  the current state.  \var{n} is a non-negative integer which is used to
  scramble the current state vector.  This is most useful in multi-threaded
  programs, in conjuction with multiple instances of the \class{Random}
  class: \method{setstate()} or \method{seed()} can be used to force all
  instances into the same internal state, and then \method{jumpahead()}
  can be used to force the instances' states far apart.
  \versionadded{2.1}
  \versionchanged[Instead of jumping to a specific state, \var{n} steps
  ahead, \method{jumpahead(\var{n})} jumps to another state likely to be
  separated by many steps]{2.3}
 \end{funcdesc}

\begin{funcdesc}{getrandbits}{k}
  Returns a python \class{long} int with \var{k} random bits.
  This method is supplied with the MersenneTwister generator and some
  other generators may also provide it as an optional part of the API.
  When available, \method{getrandbits()} enables \method{randrange()}
  to handle arbitrarily large ranges.
  \versionadded{2.4}
\end{funcdesc}

Functions for integers:

\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}{randint}{a, b}
  Return a random integer \var{N} such that
  \code{\var{a} <= \var{N} <= \var{b}}.
\end{funcdesc}


Functions for sequences:

\begin{funcdesc}{choice}{seq}
  Return a random element from the non-empty sequence \var{seq}.
  If \var{seq} is empty, raises \exception{IndexError}.
\end{funcdesc}

\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{funcdesc}{sample}{population, k}
  Return a \var{k} length list of unique elements chosen from the
  population sequence.  Used for random sampling without replacement.
  \versionadded{2.3}

  Returns a new list containing elements from the population while
  leaving the original population unchanged.  The resulting list is
  in selection order so that all sub-slices will also be valid random
  samples.  This allows raffle winners (the sample) to be partitioned
  into grand prize and second place winners (the subslices).

  Members of the population need not be hashable or unique.  If the
  population contains repeats, then each occurrence is a possible
  selection in the sample.

  To choose a sample from a range of integers, use an \function{xrange()}
  object as an argument.  This is especially fast and space efficient for
  sampling from a large population:  \code{sample(xrange(10000000), 60)}.
\end{funcdesc}


The following functions generate specific real-valued distributions.
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}{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}

\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}{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}{gammavariate}{alpha, beta}
  Gamma distribution.  (\emph{Not} the gamma function!)  Conditions on
  the parameters are \code{\var{alpha} > 0} 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}

Alternative Generators

\begin{classdesc}{WichmannHill}{\optional{seed}}
Class that implements the Wichmann-Hill algorithm as the core generator.
Has all of the same methods as \class{Random} plus the \method{whseed()}
method described below.  Because this class is implemented in pure
Python, it is not threadsafe and may require locks between calls.  The
period of the generator is 6,953,607,871,644 which is small enough to
require care that two independent random sequences do not overlap.
\end{classdesc}

\begin{funcdesc}{whseed}{\optional{x}}
  This is obsolete, supplied for bit-level compatibility with versions
  of Python prior to 2.1.
  See \function{seed()} for details.  \function{whseed()} does not guarantee
  that distinct integer arguments yield distinct internal states, and can
  yield no more than about 2**24 distinct internal states in all.
\end{funcdesc}

\begin{classdesc}{SystemRandom}{\optional{seed}}
Class that uses the \function{os.urandom()} function for generating
random numbers from sources provided by the operating system.
Not available on all systems.
Does not rely on software state and sequences are not reproducible.
Accordingly, the \method{seed()} and \method{jumpahead()} methods
have no effect and are ignored.  The \method{getstate()} and
\method{setstate()} methods raise \exception{NotImplementedError} if
called.
\versionadded{2.4}
\end{classdesc}

\begin{seealso}
  \seetext{M. Matsumoto and T. Nishimura, ``Mersenne Twister: A
	   623-dimensionally equidistributed uniform pseudorandom
	   number generator'',
	   \citetitle{ACM Transactions on Modeling and Computer Simulation}
	   Vol. 8, No. 1, January pp.3-30 1998.}

  \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.}

  \seeurl{http://www.npl.co.uk/ssfm/download/abstracts.html\#196}{A modern
          variation of the Wichmann-Hill generator that greatly increases
          the period, and passes now-standard statistical tests that the
          original generator failed.}
\end{seealso}