Show 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 The functions
supplied by this module are actually bound methods of a hidden instance of the Class
The Warning The pseudo-random generators of this module should not be used for security purposes. For security or cryptographic uses, see the
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 If a is an int, it is used directly. With version 2 (the default), a With version 1 (provided for reproducing random sequences from older versions of Python), the algorithm for Changed in version 3.2: Moved to the version 2 scheme which uses all of the bits in a string seed. Deprecated since version 3.9: In the future, the seed must be one of the following types: NoneType,
`random.` `getstate` ()¶Return an object capturing the current internal state of the generator. This object can be passed to
`random.` `setstate` (state)¶state should have been obtained from a previous call to
## Functions for bytes¶`random.` `randbytes` (n)¶Generate n random bytes. This method should not be used for
generating security tokens. Use New in version 3.9. ## Functions for integers¶`random.` `randrange` (stop)¶ `random.` `randrange` (start, stop[, step])Return a randomly selected element from The
positional argument pattern matches that of Changed in version 3.2: Deprecated since version 3.10: The automatic conversion of non-integer types to equivalent integers is deprecated. Currently Deprecated since version 3.10: The exception raised for non-integral
values such as `random.` `randint` (a,
b)¶Return a random integer N such that `random.` `getrandbits` (k)¶
Returns a non-negative 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, 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 `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 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
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 The weights or cum_weights can use any numeric type that interoperates with the For a given seed, the
New in version 3.6. Changed in version 3.9: Raises a
`random.` `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 To shuffle an immutable sequence and return a new shuffled list, use Note that even for small Deprecated since version 3.9, will be 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 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 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, To choose a sample from a range of integers, use a If the sample size is larger than the population size, a
Changed in version 3.9: Added the counts parameter. Deprecated since version 3.9: In the future, the population must be a sequence. Instances of ## 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, 1.0). `random.` `uniform` (a,
b)¶Return a random floating point number N such that The end-point value `random.` `triangular` (low, high, mode)¶Return a random floating point number N such that `random.` `betavariate` (alpha, beta)¶Beta distribution. Conditions on the parameters are `random.` `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. `random.` `gammavariate` (alpha, beta)¶Gamma distribution. (Not the gamma function!) Conditions on the parameters are The probability distribution function is: x ** (alpha - 1) * math.exp(-x / beta) pdf(x) = -------------------------------------- math.gamma(alpha) * beta ** alpha `random.` `gauss` (mu, sigma)¶Normal distribution, also called the Gaussian distribution. mu is the mean, and sigma is the standard deviation. This is slightly faster than
the 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 `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, sigma)¶Normal distribution. mu is the mean, and sigma is the standard deviation. `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
Deprecated since version 3.9: In the future, the seed must be one of the following types: `random.` `SystemRandom` ([seed])¶Class that uses the ## 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 `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. >>> dealt = sample(['tens', 'low cards'], counts=[16, 36], k=20) >>> dealt.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(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¶The default 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 |