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

.. module:: random
   :synopsis: Generate pseudo-random numbers with various common distributions.

**Source code:** :source:`Lib/random.py`

--------------

This module implements pseudo-random number generators for various
distributions.

For integers, there is uniform selection from a range. For sequences, there is
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.

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`, and :meth:`setstate` methods.
Optionally, a new generator can supply a :meth:`getrandbits` method --- this
allows :meth:`randrange` to produce selections over an arbitrarily large range.

The :mod:`random` module also provides the :class:`SystemRandom` class which
uses the system function :func:`os.urandom` to generate random numbers
from sources provided by the operating system.


Bookkeeping functions:

.. function:: seed([x], version=2)

   Initialize the random number generator.

   If *x* is omitted or ``None``, the current system time is used.  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 an int, it is used directly.

   With version 2 (the default), a :class:`str`, :class:`bytes`, or :class:`bytearray`
   object gets converted to an :class:`int` and all of its bits are used.  With version 1,
   the :func:`hash` of *x* is used instead.

   .. versionchanged:: 3.2
      Moved to the version 2 scheme which uses all of the bits in a string seed.

.. 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:: getrandbits(k)

   Returns a Python integer 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.

   The positional argument pattern matches that of :func:`range`.  Keyword arguments
   should not be used because the function may use them in unexpected ways.

   .. versionchanged:: 3.2
      :meth:`randrange` is more sophisticated about producing equally distributed
      values.  Formerly it used a style like ``int(random()*n)`` which could produce
      slightly uneven distributions.

.. function:: randint(a, b)

   Return a random integer *N* such that ``a <= N <= b``.  Alias for
   ``randrange(a, b+1)``.


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
   or set. 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 :term:`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`` for
   ``a <= b`` and ``b <= N <= a`` for ``b < a``.

   The end-point value ``b`` may or may not be included in the range
   depending on floating-point rounding in the equation ``a + (b-a) * random()``.


.. function:: triangular(low, high, mode)

   Return a random floating point number *N* such that ``low <= N <= high`` and
   with the specified *mode* between those bounds.  The *low* and *high* bounds
   default to zero and one.  The *mode* argument defaults to the midpoint
   between the bounds, giving a symmetric distribution.


.. 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.  It should be nonzero.  (The parameter would be called
   "lambda", but that is a reserved word in Python.)  Returned values
   range from 0 to positive infinity if *lambd* is positive, and from
   negative infinity to 0 if *lambd* is negative.


.. function:: gammavariate(alpha, beta)

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

   The probability distribution function is::

                 x ** (alpha - 1) * math.exp(-x / beta)
       pdf(x) =  --------------------------------------
                   math.gamma(alpha) * beta ** alpha


.. 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 Generator:

.. 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` method has no effect and is ignored.
   The :meth:`getstate` and :meth:`setstate` methods raise
   :exc:`NotImplementedError` if called.


.. 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.


   `Complementary-Multiply-with-Carry recipe
   <http://code.activestate.com/recipes/576707/>`_ for a compatible alternative
   random number generator with a long period and comparatively simple update
   operations.


Notes on Reproducibility
------------------------

Sometimes it is useful to be able to reproduce the sequences given by a pseudo
random number generator.  By re-using a seed value, the same sequence should be
reproducible from run to run as long as multiple threads are not running.

Most of the random module's algorithms and seeding functions are subject to
change across Python versions, but two aspects are guaranteed not to change:

* If a new seeding method is added, then a backward compatible seeder will be
  offered.

* The generator's :meth:`random` method will continue to produce the same
  sequence when the compatible seeder is given the same seed.

.. _random-examples:

Examples and Recipes
--------------------

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.randrange(10)                 # Integer from 0 to 9
   7

   >>> random.randrange(0, 101, 2)          # Even integer from 0 to 100
   26

   >>> random.choice('abcdefghij')          # Single 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)   # Three samples without replacement
   [4, 1, 5]

A common task is to make a :func:`random.choice` with weighted probababilites.

If the weights are small integer ratios, a simple technique is to build a sample
population with repeats::

    >>> weighted_choices = [('Red', 3), ('Blue', 2), ('Yellow', 1), ('Green', 4)]
    >>> population = [val for val, cnt in weighted_choices for i in range(cnt)]
    >>> random.choice(population)
    'Green'

A more general approach is to arrange the weights in a cumulative distribution
with :func:`itertools.accumulate`, and then locate the random value with
:func:`bisect.bisect`::

    >>> choices, weights = zip(*weighted_choices)
    >>> cumdist = list(itertools.accumulate(weights))
    >>> x = random.random() * cumdist[-1]
    >>> choices[bisect.bisect(cumdist, x)]
    'Blue'
