491 lines
18 KiB
ReStructuredText
491 lines
18 KiB
ReStructuredText
: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.random`,
|
|
:meth:`~Random.seed`, :meth:`~Random.getstate`, and :meth:`~Random.setstate` methods.
|
|
Optionally, a new generator can supply a :meth:`~Random.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.
|
|
|
|
.. warning::
|
|
|
|
The pseudo-random generators of this module should not be used for
|
|
security purposes. For security or cryptographic uses, see the
|
|
:mod:`secrets` module.
|
|
|
|
.. 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
|
|
<https://code.activestate.com/recipes/576707/>`_ for a compatible alternative
|
|
random number generator with a long period and comparatively simple update
|
|
operations.
|
|
|
|
|
|
Bookkeeping functions
|
|
---------------------
|
|
|
|
.. function:: 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 :func:`os.urandom` function for details
|
|
on availability).
|
|
|
|
If *a* 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 (provided for reproducing random sequences from older versions
|
|
of Python), the algorithm for :class:`str` and :class:`bytes` generates a
|
|
narrower range of seeds.
|
|
|
|
.. 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:`getstate` 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(stop)
|
|
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:: 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 :exc:`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 :func:`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 :exc:`TypeError`
|
|
to specify both *weights* and *cum_weights*.
|
|
|
|
The *weights* or *cum_weights* can use any numeric type that interoperates
|
|
with the :class:`float` values returned by :func:`random` (that includes
|
|
integers, floats, and fractions but excludes decimals).
|
|
|
|
For a given seed, the :func:`choices` function with equal weighting
|
|
typically produces a different sequence than repeated calls to
|
|
:func:`choice`. The algorithm used by :func:`choices` uses floating
|
|
point arithmetic for internal consistency and speed. The algorithm used
|
|
by :func:`choice` defaults to integer arithmetic with repeated selections
|
|
to avoid small biases from round-off error.
|
|
|
|
.. versionadded:: 3.6
|
|
|
|
|
|
.. 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`.
|
|
|
|
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.
|
|
|
|
|
|
.. 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 a :func:`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 :exc:`ValueError`
|
|
is raised.
|
|
|
|
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.
|
|
|
|
|
|
.. 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.
|
|
|
|
|
|
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.random` method will continue to produce the same
|
|
sequence when the compatible seeder is given the same seed.
|
|
|
|
.. _random-examples:
|
|
|
|
Examples and Recipes
|
|
--------------------
|
|
|
|
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
|
|
>>> # (a ten, jack, queen, or king).
|
|
>>> deck = collections.Counter(tens=16, low_cards=36)
|
|
>>> seen = sample(list(deck.elements()), k=20)
|
|
>>> seen.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.
|
|
>>> def trial():
|
|
... return choices('HT', cum_weights=(0.60, 1.00), k=7).count('H') >= 5
|
|
...
|
|
>>> sum(trial() for i in range(10000)) / 10000
|
|
0.4169
|
|
|
|
>>> # Probability of the median of 5 samples being in middle two quartiles
|
|
>>> def trial():
|
|
... return 2500 <= sorted(choices(range(10000), k=5))[2] < 7500
|
|
...
|
|
>>> sum(trial() for i in range(10000)) / 10000
|
|
0.7958
|
|
|
|
Example of `statistical bootstrapping
|
|
<https://en.wikipedia.org/wiki/Bootstrapping_(statistics)>`_ using resampling
|
|
with replacement to estimate a confidence interval for the mean of a sample of
|
|
size five::
|
|
|
|
# http://statistics.about.com/od/Applications/a/Example-Of-Bootstrapping.htm
|
|
from statistics import mean
|
|
from random import choices
|
|
|
|
data = 1, 2, 4, 4, 10
|
|
means = sorted(mean(choices(data, k=5)) for i in range(20))
|
|
print(f'The sample mean of {mean(data):.1f} has a 90% confidence '
|
|
f'interval from {means[1]:.1f} to {means[-2]:.1f}')
|
|
|
|
Example of a `resampling permutation test
|
|
<https://en.wikipedia.org/wiki/Resampling_(statistics)#Permutation_tests>`_
|
|
to determine the statistical significance or `p-value
|
|
<https://en.wikipedia.org/wiki/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 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 = 10000
|
|
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 in a single server queue::
|
|
|
|
from random import expovariate, gauss
|
|
from statistics import mean, median, stdev
|
|
|
|
average_arrival_interval = 5.6
|
|
average_service_time = 5.0
|
|
stdev_service_time = 0.5
|
|
|
|
num_waiting = 0
|
|
arrivals = []
|
|
starts = []
|
|
arrival = service_end = 0.0
|
|
for i in range(20000):
|
|
if arrival <= service_end:
|
|
num_waiting += 1
|
|
arrival += expovariate(1.0 / average_arrival_interval)
|
|
arrivals.append(arrival)
|
|
else:
|
|
num_waiting -= 1
|
|
service_start = service_end if num_waiting else arrival
|
|
service_time = gauss(average_service_time, stdev_service_time)
|
|
service_end = service_start + service_time
|
|
starts.append(service_start)
|
|
|
|
waits = [start - arrival for arrival, start in zip(arrivals, starts)]
|
|
print(f'Mean wait: {mean(waits):.1f}. Stdev wait: {stdev(waits):.1f}.')
|
|
print(f'Median wait: {median(waits):.1f}. Max wait: {max(waits):.1f}.')
|
|
|
|
.. seealso::
|
|
|
|
`Statistics for Hackers <https://www.youtube.com/watch?v=Iq9DzN6mvYA>`_
|
|
a video tutorial by
|
|
`Jake Vanderplas <https://us.pycon.org/2016/speaker/profile/295/>`_
|
|
on statistical analysis using just a few fundamental concepts
|
|
including simulation, sampling, shuffling, and cross-validation.
|
|
|
|
`Economics Simulation
|
|
<http://nbviewer.jupyter.org/url/norvig.com/ipython/Economics.ipynb>`_
|
|
a simulation of a marketplace by
|
|
`Peter Norvig <http://norvig.com/bio.html>`_ 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)
|
|
<http://nbviewer.jupyter.org/url/norvig.com/ipython/Probability.ipynb>`_
|
|
a tutorial by `Peter Norvig <http://norvig.com/bio.html>`_ covering
|
|
the basics of probability theory, how to write simulations, and
|
|
how to perform data analysis using Python.
|