:mod:`random` --- Generate pseudo-random numbers
================================================

.. module:: random
   :synopsis: 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 :func:`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 :meth:`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 :meth:`random`,
:meth:`seed`, :meth:`getstate`, :meth:`setstate` and :meth:`jumpahead` methods.
Optionally, a new generator can supply a :meth:`getrandombits` method --- this
allows :meth:`randrange` to produce selections over an arbitrarily large range.

As an example of subclassing, the :mod:`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.

Bookkeeping functions:


.. function:: seed([x])

   Initialize the basic random number generator. Optional argument *x* can be any
   hashable object. If *x* is omitted or ``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 :func:`os.urandom` function
   for details on availability).

   If *x* is not ``None`` or an int or long, ``hash(x)`` is used instead. If *x* is
   an int or long, *x* is used directly.


.. function:: getstate()

   Return an object capturing the current internal state of the generator.  This
   object can be passed to :func:`setstate` to restore the state.


.. function:: setstate(state)

   *state* should have been obtained from a previous call to :func:`getstate`, and
   :func:`setstate` restores the internal state of the generator to what it was at
   the time :func:`setstate` was called.


.. function:: jumpahead(n)

   Change the internal state to one different from and likely far away from the
   current state.  *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:
   :meth:`setstate` or :meth:`seed` can be used to force all instances into the
   same internal state, and then :meth:`jumpahead` can be used to force the
   instances' states far apart.


.. function:: getrandbits(k)

   Returns a python :class:`long` int with *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, :meth:`getrandbits` enables
   :meth:`randrange` to handle arbitrarily large ranges.


Functions for integers:

.. function:: randrange([start,] stop[, step])

   Return a randomly selected element from ``range(start, stop, step)``.  This is
   equivalent to ``choice(range(start, stop, step))``, but doesn't actually build a
   range object.


.. function:: randint(a, b)

   Return a random integer *N* such that ``a <= N <= b``.


Functions for sequences:

.. function:: choice(seq)

   Return a random element from the non-empty sequence *seq*. If *seq* is empty,
   raises :exc:`IndexError`.


.. function:: shuffle(x[, random])

   Shuffle the sequence *x* in place. The optional argument *random* is a
   0-argument function returning a random float in [0.0, 1.0); by default, this is
   the function :func:`random`.

   Note that for even rather small ``len(x)``, the total number of permutations of
   *x* is larger than the period of most random number generators; this implies
   that most permutations of a long sequence can never be generated.


.. function:: sample(population, k)

   Return a *k* length list of unique elements chosen from the population sequence.
   Used for random sampling without replacement.

   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 :func:`range` object as an
   argument.  This is especially fast and space efficient for sampling from a large
   population:  ``sample(range(10000000), 60)``.

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.


.. function:: random()

   Return the next random floating point number in the range [0.0, 1.0).


.. function:: uniform(a, b)

   Return a random floating point number *N* such that ``a <= N < b``.


.. function:: betavariate(alpha, beta)

   Beta distribution.  Conditions on the parameters are ``alpha > 0`` and ``beta >
   0``. Returned values range between 0 and 1.


.. function:: expovariate(lambd)

   Exponential distribution.  *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.


.. function:: gammavariate(alpha, beta)

   Gamma distribution.  (*Not* the gamma function!)  Conditions on the parameters
   are ``alpha > 0`` and ``beta > 0``.


.. function:: gauss(mu, sigma)

   Gaussian distribution.  *mu* is the mean, and *sigma* is the standard deviation.
   This is slightly faster than the :func:`normalvariate` function defined below.


.. function:: lognormvariate(mu, sigma)

   Log normal distribution.  If you take the natural logarithm of this
   distribution, you'll get a normal distribution with mean *mu* and standard
   deviation *sigma*.  *mu* can have any value, and *sigma* must be greater than
   zero.


.. function:: normalvariate(mu, sigma)

   Normal distribution.  *mu* is the mean, and *sigma* is the standard deviation.


.. function:: vonmisesvariate(mu, kappa)

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


.. function:: paretovariate(alpha)

   Pareto distribution.  *alpha* is the shape parameter.


.. function:: weibullvariate(alpha, beta)

   Weibull distribution.  *alpha* is the scale parameter and *beta* is the shape
   parameter.


Alternative Generators:

.. class:: WichmannHill([seed])

   Class that implements the Wichmann-Hill algorithm as the core generator. Has all
   of the same methods as :class:`Random` plus the :meth:`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.


.. function:: whseed([x])

   This is obsolete, supplied for bit-level compatibility with versions of Python
   prior to 2.1. See :func:`seed` for details.  :func:`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.


.. class:: SystemRandom([seed])

   Class that uses the :func:`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 :meth:`seed` and :meth:`jumpahead` methods have no effect and are ignored.
   The :meth:`getstate` and :meth:`setstate` methods raise
   :exc:`NotImplementedError` if called.


Examples of basic usage::

   >>> random.random()        # Random float x, 0.0 <= x < 1.0
   0.37444887175646646
   >>> random.uniform(1, 10)  # Random float x, 1.0 <= x < 10.0
   1.1800146073117523
   >>> random.randint(1, 10)  # Integer from 1 to 10, endpoints included
   7
   >>> random.randrange(0, 101, 2)  # Even integer from 0 to 100
   26
   >>> random.choice('abcdefghij')  # Choose a random element
   'c'

   >>> items = [1, 2, 3, 4, 5, 6, 7]
   >>> random.shuffle(items)
   >>> items
   [7, 3, 2, 5, 6, 4, 1]

   >>> random.sample([1, 2, 3, 4, 5],  3)  # Choose 3 elements
   [4, 1, 5]



.. seealso::

   M. Matsumoto and T. Nishimura, "Mersenne Twister: A 623-dimensionally
   equidistributed uniform pseudorandom number generator", ACM Transactions on
   Modeling and Computer Simulation Vol. 8, No. 1, January pp.3-30 1998.

   Wichmann, B. A. & Hill, I. D., "Algorithm AS 183: An efficient and portable
   pseudo-random number generator", Applied Statistics 31 (1982) 188-190.

   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.