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abc.rst.txt	11805	0644	editdlrm
aifc.rst.txt	7503	0644	editdlrm
allos.rst.txt	678	0644	editdlrm
archiving.rst.txt	440	0644	editdlrm
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ast.rst.txt	10159	0644	editdlrm
asynchat.rst.txt	8528	0644	editdlrm
asyncio-dev.rst.txt	13398	0644	editdlrm
asyncio-eventloop.rst.txt	33701	0644	editdlrm
asyncio-eventloops.rst.txt	7660	0644	editdlrm
asyncio-protocol.rst.txt	25170	0644	editdlrm
asyncio-queue.rst.txt	4328	0644	editdlrm
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asyncio-sync.rst.txt	9078	0644	editdlrm
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asyncio.rst.txt	2272	0644	editdlrm
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atexit.rst.txt	3669	0644	editdlrm
audioop.rst.txt	10712	0644	editdlrm
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builtins.rst.txt	1465	0644	editdlrm
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collections.abc.rst.txt	12861	0644	editdlrm
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crypt.rst.txt	5122	0644	editdlrm
crypto.rst.txt	411	0644	editdlrm
csv.rst.txt	19902	0644	editdlrm
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datetime.rst.txt	90524	0644	editdlrm
dbm.rst.txt	14293	0644	editdlrm
debug.rst.txt	471	0644	editdlrm
decimal.rst.txt	74869	0644	editdlrm
development.rst.txt	704	0644	editdlrm
difflib.rst.txt	29861	0644	editdlrm
dis.rst.txt	32567	0644	editdlrm
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doctest.rst.txt	71743	0644	editdlrm
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faulthandler.rst.txt	6136	0644	editdlrm
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fnmatch.rst.txt	3103	0644	editdlrm
formatter.rst.txt	13242	0644	editdlrm
fpectl.rst.txt	4180	0644	editdlrm
fractions.rst.txt	6364	0644	editdlrm
frameworks.rst.txt	391	0644	editdlrm
ftplib.rst.txt	17552	0644	editdlrm
functional.rst.txt	365	0644	editdlrm
functions.rst.txt	71621	0644	editdlrm
functools.rst.txt	18503	0644	editdlrm
gc.rst.txt	9694	0644	editdlrm
getopt.rst.txt	6557	0644	editdlrm
getpass.rst.txt	1882	0644	editdlrm
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hashlib.rst.txt	26523	0644	editdlrm
heapq.rst.txt	13402	0644	editdlrm
hmac.rst.txt	4020	0644	editdlrm
html.entities.rst.txt	1319	0644	editdlrm
html.parser.rst.txt	11275	0644	editdlrm
html.rst.txt	1304	0644	editdlrm
http.client.rst.txt	18870	0644	editdlrm
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http.rst.txt	7096	0644	editdlrm
http.server.rst.txt	17639	0644	editdlrm
i18n.rst.txt	408	0644	editdlrm
idle.rst.txt	25834	0644	editdlrm
imaplib.rst.txt	20367	0644	editdlrm
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imp.rst.txt	15548	0644	editdlrm
importlib.rst.txt	53563	0644	editdlrm
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ipc.rst.txt	649	0644	editdlrm
itertools.rst.txt	38459	0644	editdlrm
json.rst.txt	27876	0644	editdlrm
keyword.rst.txt	617	0644	editdlrm
language.rst.txt	522	0644	editdlrm
linecache.rst.txt	2408	0644	editdlrm
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logging.config.rst.txt	33886	0644	editdlrm
logging.handlers.rst.txt	42497	0644	editdlrm
logging.rst.txt	60479	0644	editdlrm
lzma.rst.txt	17361	0644	editdlrm
macpath.rst.txt	707	0644	editdlrm
mailbox.rst.txt	63055	0644	editdlrm
mailcap.rst.txt	3691	0644	editdlrm
markup.rst.txt	679	0644	editdlrm
marshal.rst.txt	4855	0644	editdlrm
math.rst.txt	14375	0644	editdlrm
mimetypes.rst.txt	9839	0644	editdlrm
misc.rst.txt	247	0644	editdlrm
mm.rst.txt	431	0644	editdlrm
mmap.rst.txt	11232	0644	editdlrm
modulefinder.rst.txt	3240	0644	editdlrm
modules.rst.txt	355	0644	editdlrm
msilib.rst.txt	18612	0644	editdlrm
msvcrt.rst.txt	4392	0644	editdlrm
multiprocessing.rst.txt	104494	0644	editdlrm
netdata.rst.txt	339	0644	editdlrm
netrc.rst.txt	3007	0644	editdlrm
nis.rst.txt	1993	0644	editdlrm
nntplib.rst.txt	21821	0644	editdlrm
numbers.rst.txt	8089	0644	editdlrm
numeric.rst.txt	696	0644	editdlrm
operator.rst.txt	18989	0644	editdlrm
optparse.rst.txt	76996	0644	editdlrm
os.path.rst.txt	16419	0644	editdlrm
os.rst.txt	137534	0644	editdlrm
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othergui.rst.txt	2822	0644	editdlrm
parser.rst.txt	15148	0644	editdlrm
pathlib.rst.txt	30045	0644	editdlrm
pdb.rst.txt	19454	0644	editdlrm
persistence.rst.txt	591	0644	editdlrm
pickle.rst.txt	37440	0644	editdlrm
pickletools.rst.txt	3729	0644	editdlrm
pipes.rst.txt	2557	0644	editdlrm
pkgutil.rst.txt	8730	0644	editdlrm
platform.rst.txt	9624	0644	editdlrm
plistlib.rst.txt	7398	0644	editdlrm
poplib.rst.txt	8197	0644	editdlrm
posix.rst.txt	3694	0644	editdlrm
pprint.rst.txt	14274	0644	editdlrm
profile.rst.txt	27969	0644	editdlrm
pty.rst.txt	3123	0644	editdlrm
pwd.rst.txt	2739	0644	editdlrm
pyclbr.rst.txt	3297	0644	editdlrm
pydoc.rst.txt	4688	0644	editdlrm
pyexpat.rst.txt	28620	0644	editdlrm
python.rst.txt	475	0644	editdlrm
py_compile.rst.txt	3851	0644	editdlrm
queue.rst.txt	7268	0644	editdlrm
quopri.rst.txt	2571	0644	editdlrm
random.rst.txt	18503	0644	editdlrm
re.rst.txt	64936	0644	editdlrm
readline.rst.txt	11943	0644	editdlrm
reprlib.rst.txt	5162	0644	editdlrm
resource.rst.txt	12291	0644	editdlrm
rlcompleter.rst.txt	2294	0644	editdlrm
runpy.rst.txt	8285	0644	editdlrm
sched.rst.txt	4842	0644	editdlrm
secrets.rst.txt	5930	0644	editdlrm
select.rst.txt	28175	0644	editdlrm
selectors.rst.txt	8925	0644	editdlrm
shelve.rst.txt	8390	0644	editdlrm
shlex.rst.txt	16441	0644	editdlrm
shutil.rst.txt	26014	0644	editdlrm
signal.rst.txt	17218	0644	editdlrm
site.rst.txt	9628	0644	editdlrm
smtpd.rst.txt	10806	0644	editdlrm
smtplib.rst.txt	23439	0644	editdlrm
sndhdr.rst.txt	1994	0644	editdlrm
socket.rst.txt	68202	0644	editdlrm
socketserver.rst.txt	23712	0644	editdlrm
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sqlite3.rst.txt	39215	0644	editdlrm
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stat.rst.txt	9805	0644	editdlrm
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stdtypes.rst.txt	179240	0644	editdlrm
string.rst.txt	35374	0644	editdlrm
stringprep.rst.txt	4279	0644	editdlrm
struct.rst.txt	19581	0644	editdlrm
subprocess.rst.txt	45793	0644	editdlrm
sunau.rst.txt	7390	0644	editdlrm
superseded.rst.txt	258	0644	editdlrm
symbol.rst.txt	975	0644	editdlrm
symtable.rst.txt	4964	0644	editdlrm
sys.rst.txt	58273	0644	editdlrm
sysconfig.rst.txt	8749	0644	editdlrm
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tabnanny.rst.txt	2007	0644	editdlrm
tarfile.rst.txt	31749	0644	editdlrm
telnetlib.rst.txt	7899	0644	editdlrm
tempfile.rst.txt	13857	0644	editdlrm
termios.rst.txt	3751	0644	editdlrm
test.rst.txt	25764	0644	editdlrm
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__future__.rst.txt	5245	0644	editdlrm
__main__.rst.txt	904	0644	editdlrm

: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 `_ 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). .. 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. >>> trial = lambda: 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 >>> trial = lambda : 2500 <= sorted(choices(range(10000), k=5))[2] < 7500 >>> sum(trial() for i in range(10000)) / 10000 0.7958 Example of `statistical bootstrapping `_ 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 `_ 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 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 `_ 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.