: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:`getrandbits` method --- this

allows :meth:`randrange` to produce selections over an arbitrarily large range.

.. versionadded:: 2.4

   the :meth:`getrandbits` method.

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.

.. versionchanged:: 2.3

   Substituted MersenneTwister for Wichmann-Hill.

Bookkeeping functions:

.. function:: seed([x])

   Initialize the basic random number generator. Optional argument *x* can be any

   :term:`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).

   .. versionchanged:: 2.4

      formerly, operating system resources were not used.

   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.

   .. versionadded:: 2.1

   .. versionchanged:: 2.6

      State values produced in Python 2.6 cannot be loaded into earlier versions.

.. 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.

   .. versionadded:: 2.1

.. 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

   conjunction 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.

   .. versionadded:: 2.1

   .. versionchanged:: 2.3

      Instead of jumping to a specific state, *n* steps ahead, ``jumpahead(n)``

      jumps to another state likely to be separated by many steps.

.. 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.

   .. versionadded:: 2.4

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.

   .. versionadded:: 1.5.2

.. 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.

   .. 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 :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:`xrange` object as an

   argument.  This is especially fast and space efficient for sampling from a large

   population:  ``sample(xrange(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``.

.. 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.

   .. versionadded:: 2.6

.. 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.

   .. versionadded:: 2.4

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.