"random" --- Generate pseudo-random numbers
*******************************************

**Source code:** 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 "random()",
which generates a random float uniformly in the half-open range "0.0
<= X < 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 "random.Random" class.  You can instantiate
your own instances of "Random" to get generators that don't share
state.

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

The "random" module also provides the "SystemRandom" class which uses
the system function "os.urandom()" to generate random numbers from
sources provided by the operating system.

Warning:

  The pseudo-random generators of this module should not be used for
  security purposes.  For security or cryptographic uses, see the
  "secrets" module.

See also:

  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 for a compatible
  alternative random number generator with a long period and
  comparatively simple update operations.


Bookkeeping functions
=====================

random.seed(a=None, version=2)

   Initialize the random number generator.

   If *a* 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 "os.urandom()" function
   for details on availability).

   If *a* is an int, it is used directly.

   With version 2 (the default), a "str", "bytes", or "bytearray"
   object gets converted to an "int" and all of its bits are used.

   With version 1 (provided for reproducing random sequences from
   older versions of Python), the algorithm for "str" and "bytes"
   generates a narrower range of seeds.

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

   Changed in version 3.11: The *seed* must be one of the following
   types: *NoneType*, "int", "float", "str", "bytes", or "bytearray".

random.getstate()

   Return an object capturing the current internal state of the
   generator.  This object can be passed to "setstate()" to restore
   the state.

random.setstate(state)

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


Functions for bytes
===================

random.randbytes(n)

   Generate *n* random bytes.

   This method should not be used for generating security tokens. Use
   "secrets.token_bytes()" instead.

   New in version 3.9.


Functions for integers
======================

random.randrange(stop)
random.randrange(start, stop[, step])

   Return a randomly selected element from "range(start, stop, step)".

   This is roughly equivalent to "choice(range(start, stop, step))"
   but supports arbitrarily large ranges and is optimized for common
   cases.

   The positional argument pattern matches the "range()" function.

   Keyword arguments should not be used because they can be
   interpreted in unexpected ways. For example "randrange(start=100)"
   is interpreted as "randrange(0, 100, 1)".

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

   Changed in version 3.12: Automatic conversion of non-integer types
   is no longer supported. Calls such as "randrange(10.0)" and
   "randrange(Fraction(10, 1))" now raise a "TypeError".

random.randint(a, b)

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

random.getrandbits(k)

   Returns a non-negative Python integer with *k* random bits. This
   method is supplied with the Mersenne Twister generator and some
   other generators may also provide it as an optional part of the
   API. When available, "getrandbits()" enables "randrange()" to
   handle arbitrarily large ranges.

   Changed in version 3.9: This method now accepts zero for *k*.


Functions for sequences
=======================

random.choice(seq)

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

random.choices(population, weights=None, *, cum_weights=None, k=1)

   Return a *k* sized list of elements chosen from the *population*
   with replacement. If the *population* is empty, raises
   "IndexError".

   If a *weights* sequence is specified, selections are made according
   to the relative weights.  Alternatively, if a *cum_weights*
   sequence is given, the selections are made according to the
   cumulative weights (perhaps computed using
   "itertools.accumulate()").  For example, the relative weights "[10,
   5, 30, 5]" are equivalent to the cumulative weights "[10, 15, 45,
   50]".  Internally, the relative weights are converted to cumulative
   weights before making selections, so supplying the cumulative
   weights saves work.

   If neither *weights* nor *cum_weights* are specified, selections
   are made with equal probability.  If a weights sequence is
   supplied, it must be the same length as the *population* sequence.
   It is a "TypeError" to specify both *weights* and *cum_weights*.

   The *weights* or *cum_weights* can use any numeric type that
   interoperates with the "float" values returned by "random()" (that
   includes integers, floats, and fractions but excludes decimals).
   Weights are assumed to be non-negative and finite.  A "ValueError"
   is raised if all weights are zero.

   For a given seed, the "choices()" function with equal weighting
   typically produces a different sequence than repeated calls to
   "choice()".  The algorithm used by "choices()" uses floating point
   arithmetic for internal consistency and speed.  The algorithm used
   by "choice()" defaults to integer arithmetic with repeated
   selections to avoid small biases from round-off error.

   New in version 3.6.

   Changed in version 3.9: Raises a "ValueError" if all weights are
   zero.

random.shuffle(x)

   Shuffle the sequence *x* in place.

   To shuffle an immutable sequence and return a new shuffled list,
   use "sample(x, k=len(x))" instead.

   Note that even for small "len(x)", the total number of permutations
   of *x* can quickly grow larger than the period of most random
   number generators. This implies that most permutations of a long
   sequence can never be generated.  For example, a sequence of length
   2080 is the largest that can fit within the period of the Mersenne
   Twister random number generator.

   Deprecated since version 3.9, removed in version 3.11: The optional
   parameter *random*.

random.sample(population, k, *, counts=None)

   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.

   Repeated elements can be specified one at a time or with the
   optional keyword-only *counts* parameter.  For example,
   "sample(['red', 'blue'], counts=[4, 2], k=5)" is equivalent to
   "sample(['red', 'red', 'red', 'red', 'blue', 'blue'], k=5)".

   To choose a sample from a range of integers, use a "range()" object
   as an argument.  This is especially fast and space efficient for
   sampling from a large population:  "sample(range(10000000), k=60)".

   If the sample size is larger than the population size, a
   "ValueError" is raised.

   Changed in version 3.9: Added the *counts* parameter.

   Changed in version 3.11: The *population* must be a sequence.
   Automatic conversion of sets to lists is no longer supported.


Discrete distributions
======================

The following function generates a discrete distribution.

random.binomialvariate(n=1, p=0.5)

   Binomial distribution. Return the number of successes for *n*
   independent trials with the probability of success in each trial
   being *p*:

   Mathematically equivalent to:

      sum(random() < p for i in range(n))

   The number of trials *n* should be a non-negative integer. The
   probability of success *p* should be between "0.0 <= p <= 1.0". The
   result is an integer in the range "0 <= X <= n".

   New in version 3.12.


Real-valued distributions
=========================

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.

random.random()

   Return the next random floating point number in the range "0.0 <= X
   < 1.0"

random.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()".

random.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.

random.betavariate(alpha, beta)

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

random.expovariate(lambd=1.0)

   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.

   Changed in version 3.12: Added the default value for "lambd".

random.gammavariate(alpha, beta)

   Gamma distribution.  (*Not* the gamma function!)  The shape and
   scale parameters, *alpha* and *beta*, must have positive values.
   (Calling conventions vary and some sources define 'beta' as the
   inverse of the scale).

   The probability distribution function is:

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

random.gauss(mu=0.0, sigma=1.0)

   Normal distribution, also called the Gaussian distribution. *mu* is
   the mean, and *sigma* is the standard deviation.  This is slightly
   faster than the "normalvariate()" function defined below.

   Multithreading note:  When two threads call this function
   simultaneously, it is possible that they will receive the same
   return value.  This can be avoided in three ways. 1) Have each
   thread use a different instance of the random number generator. 2)
   Put locks around all calls. 3) Use the slower, but thread-safe
   "normalvariate()" function instead.

   Changed in version 3.11: *mu* and *sigma* now have default
   arguments.

random.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.

random.normalvariate(mu=0.0, sigma=1.0)

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

   Changed in version 3.11: *mu* and *sigma* now have default
   arguments.

random.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*.

random.paretovariate(alpha)

   Pareto distribution.  *alpha* is the shape parameter.

random.weibullvariate(alpha, beta)

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


Alternative Generator
=====================

class random.Random([seed])

   Class that implements the default pseudo-random number generator
   used by the "random" module.

   Deprecated since version 3.9, removed in version 3.11: Formerly the
   *seed* could be any hashable object.  Now it is limited to:
   "NoneType", "int", "float", "str", "bytes", or "bytearray".

class random.SystemRandom([seed])

   Class that uses the "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 "seed()" method
   has no effect and is ignored. The "getstate()" and "setstate()"
   methods raise "NotImplementedError" if called.


Notes on Reproducibility
========================

Sometimes it is useful to be able to reproduce the sequences given by
a pseudo-random number generator.  By reusing 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 "random()" method will continue to produce the same
  sequence when the compatible seeder is given the same seed.


Examples
========

Basic examples:

   >>> random()                             # Random float:  0.0 <= x < 1.0
   0.37444887175646646

   >>> uniform(2.5, 10.0)                   # Random float:  2.5 <= x <= 10.0
   3.1800146073117523

   >>> expovariate(1 / 5)                   # Interval between arrivals averaging 5 seconds
   5.148957571865031

   >>> randrange(10)                        # Integer from 0 to 9 inclusive
   7

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

   >>> choice(['win', 'lose', 'draw'])      # Single random element from a sequence
   'draw'

   >>> deck = 'ace two three four'.split()
   >>> shuffle(deck)                        # Shuffle a list
   >>> deck
   ['four', 'two', 'ace', 'three']

   >>> sample([10, 20, 30, 40, 50], k=4)    # Four samples without replacement
   [40, 10, 50, 30]

Simulations:

   >>> # Six roulette wheel spins (weighted sampling with replacement)
   >>> choices(['red', 'black', 'green'], [18, 18, 2], k=6)
   ['red', 'green', 'black', 'black', 'red', 'black']

   >>> # Deal 20 cards without replacement from a deck
   >>> # of 52 playing cards, and determine the proportion of cards
   >>> # with a ten-value:  ten, jack, queen, or king.
   >>> deal = sample(['tens', 'low cards'], counts=[16, 36], k=20)
   >>> deal.count('tens') / 20
   0.15

   >>> # Estimate the probability of getting 5 or more heads from 7 spins
   >>> # of a biased coin that settles on heads 60% of the time.
   >>> sum(binomialvariate(n=7, p=0.6) >= 5 for i in range(10_000)) / 10_000
   0.4169

   >>> # Probability of the median of 5 samples being in middle two quartiles
   >>> def trial():
   ...     return 2_500 <= sorted(choices(range(10_000), k=5))[2] < 7_500
   ...
   >>> sum(trial() for i in range(10_000)) / 10_000
   0.7958

Example of statistical bootstrapping using resampling with replacement
to estimate a confidence interval for the mean of a sample:

   # https://www.thoughtco.com/example-of-bootstrapping-3126155
   from statistics import fmean as mean
   from random import choices

   data = [41, 50, 29, 37, 81, 30, 73, 63, 20, 35, 68, 22, 60, 31, 95]
   means = sorted(mean(choices(data, k=len(data))) for i in range(100))
   print(f'The sample mean of {mean(data):.1f} has a 90% confidence '
         f'interval from {means[5]:.1f} to {means[94]:.1f}')

Example of a resampling permutation test to determine the statistical
significance or p-value of an observed difference between the effects
of a drug versus a placebo:

   # Example from "Statistics is Easy" by Dennis Shasha and Manda Wilson
   from statistics import fmean as mean
   from random import shuffle

   drug = [54, 73, 53, 70, 73, 68, 52, 65, 65]
   placebo = [54, 51, 58, 44, 55, 52, 42, 47, 58, 46]
   observed_diff = mean(drug) - mean(placebo)

   n = 10_000
   count = 0
   combined = drug + placebo
   for i in range(n):
       shuffle(combined)
       new_diff = mean(combined[:len(drug)]) - mean(combined[len(drug):])
       count += (new_diff >= observed_diff)

   print(f'{n} label reshufflings produced only {count} instances with a difference')
   print(f'at least as extreme as the observed difference of {observed_diff:.1f}.')
   print(f'The one-sided p-value of {count / n:.4f} leads us to reject the null')
   print(f'hypothesis that there is no difference between the drug and the placebo.')

Simulation of arrival times and service deliveries for a multiserver
queue:

   from heapq import heapify, heapreplace
   from random import expovariate, gauss
   from statistics import mean, quantiles

   average_arrival_interval = 5.6
   average_service_time = 15.0
   stdev_service_time = 3.5
   num_servers = 3

   waits = []
   arrival_time = 0.0
   servers = [0.0] * num_servers  # time when each server becomes available
   heapify(servers)
   for i in range(1_000_000):
       arrival_time += expovariate(1.0 / average_arrival_interval)
       next_server_available = servers[0]
       wait = max(0.0, next_server_available - arrival_time)
       waits.append(wait)
       service_duration = max(0.0, gauss(average_service_time, stdev_service_time))
       service_completed = arrival_time + wait + service_duration
       heapreplace(servers, service_completed)

   print(f'Mean wait: {mean(waits):.1f}   Max wait: {max(waits):.1f}')
   print('Quartiles:', [round(q, 1) for q in quantiles(waits)])

See also:

  Statistics for Hackers a video tutorial by Jake Vanderplas on
  statistical analysis using just a few fundamental concepts including
  simulation, sampling, shuffling, and cross-validation.

  Economics Simulation a simulation of a marketplace by Peter Norvig
  that shows effective use of many of the tools and distributions
  provided by this module (gauss, uniform, sample, betavariate,
  choice, triangular, and randrange).

  A Concrete Introduction to Probability (using Python) a tutorial by
  Peter Norvig covering the basics of probability theory, how to write
  simulations, and how to perform data analysis using Python.


Recipes
=======

These recipes show how to efficiently make random selections from the
combinatoric iterators in the "itertools" module:

   def random_product(*args, repeat=1):
       "Random selection from itertools.product(*args, **kwds)"
       pools = [tuple(pool) for pool in args] * repeat
       return tuple(map(random.choice, pools))

   def random_permutation(iterable, r=None):
       "Random selection from itertools.permutations(iterable, r)"
       pool = tuple(iterable)
       r = len(pool) if r is None else r
       return tuple(random.sample(pool, r))

   def random_combination(iterable, r):
       "Random selection from itertools.combinations(iterable, r)"
       pool = tuple(iterable)
       n = len(pool)
       indices = sorted(random.sample(range(n), r))
       return tuple(pool[i] for i in indices)

   def random_combination_with_replacement(iterable, r):
       "Choose r elements with replacement.  Order the result to match the iterable."
       # Result will be in set(itertools.combinations_with_replacement(iterable, r)).
       pool = tuple(iterable)
       n = len(pool)
       indices = sorted(random.choices(range(n), k=r))
       return tuple(pool[i] for i in indices)

The default "random()" returns multiples of 2⁻⁵³ in the range *0.0 ≤ x
< 1.0*.  All such numbers are evenly spaced and are exactly
representable as Python floats.  However, many other representable
floats in that interval are not possible selections.  For example,
"0.05954861408025609" isn't an integer multiple of 2⁻⁵³.

The following recipe takes a different approach.  All floats in the
interval are possible selections.  The mantissa comes from a uniform
distribution of integers in the range *2⁵² ≤ mantissa < 2⁵³*.  The
exponent comes from a geometric distribution where exponents smaller
than *-53* occur half as often as the next larger exponent.

   from random import Random
   from math import ldexp

   class FullRandom(Random):

       def random(self):
           mantissa = 0x10_0000_0000_0000 | self.getrandbits(52)
           exponent = -53
           x = 0
           while not x:
               x = self.getrandbits(32)
               exponent += x.bit_length() - 32
           return ldexp(mantissa, exponent)

All real valued distributions in the class will use the new method:

   >>> fr = FullRandom()
   >>> fr.random()
   0.05954861408025609
   >>> fr.expovariate(0.25)
   8.87925541791544

The recipe is conceptually equivalent to an algorithm that chooses
from all the multiples of 2⁻¹⁰⁷⁴ in the range *0.0 ≤ x < 1.0*.  All
such numbers are evenly spaced, but most have to be rounded down to
the nearest representable Python float.  (The value 2⁻¹⁰⁷⁴ is the
smallest positive unnormalized float and is equal to "math.ulp(0.0)".)

See also:

  Generating Pseudo-random Floating-Point Values a paper by Allen B.
  Downey describing ways to generate more fine-grained floats than
  normally generated by "random()".
