9.6. "random" — Generate pseudo-random numbers
**********************************************

**Source code:** Lib/random.py

======================================================================

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 "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 "random.Random" class.  You can instantiate
your own instances of "Random" to get generators that don’t share
state.  This is especially useful for multi-threaded programs,
creating a different instance of "Random" for each thread, and using
the "jumpahead()" method to make it likely that the generated
sequences seen by each thread don’t overlap.

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()", "setstate()" and "jumpahead()"
methods.  Optionally, a new generator can supply a "getrandbits()"
method — this allows "randrange()" to produce selections over an
arbitrarily large range.

New in version 2.4: the "getrandbits()" method.

As an example of subclassing, the "random" module provides the
"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.

Changed in version 2.3: MersenneTwister replaced Wichmann-Hill as the
default generator.

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.  Use "os.urandom()" or "SystemRandom" if
  you require a cryptographically secure pseudo-random number
  generator.

Bookkeeping functions:

random.seed(a=None)

   Initialize internal state of the random number generator.

   "None" or no argument seeds from current time or from an operating
   system specific randomness source if available (see the
   "os.urandom()" function for details on availability).

   If *a* is not "None" or an "int" or a "long", then "hash(a)" is
   used instead.  Note that the hash values for some types are
   nondeterministic when "PYTHONHASHSEED" is enabled.

   Changed in version 2.4: formerly, operating system resources were
   not used.

random.getstate()

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

   New in version 2.1.

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

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.

   New in version 2.1.

random.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 "Random" class: "setstate()" or "seed()" can be used to force
   all instances into the same internal state, and then "jumpahead()"
   can be used to force the instances’ states far apart.

   New in version 2.1.

   Changed in version 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.

random.getrandbits(k)

   Returns a python "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, "getrandbits()" enables "randrange()" to handle
   arbitrarily large ranges.

   New in version 2.4.

Functions for integers:

random.randrange(stop)
random.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.

   New in version 1.5.2.

random.randint(a, b)

   Return a random integer *N* such that "a <= N <= b".

Functions for sequences:

random.choice(seq)

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

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 "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.

random.sample(population, k)

   Return a *k* length list of unique elements chosen from the
   population sequence. Used for random sampling without replacement.

   New in version 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 *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 "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.

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 "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.

   New in version 2.6.

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)

   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 "alpha > 0" and "beta > 0".

   The probability distribution function is:

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

random.gauss(mu, sigma)

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

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 Generators:

class random.WichmannHill([seed])

   Class that implements the Wichmann-Hill algorithm as the core
   generator. Has all of the same methods as "Random" plus the
   "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.

random.whseed([x])

   This is obsolete, supplied for bit-level compatibility with
   versions of Python prior to 2.1. See "seed()" for details.
   "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 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()" and
   "jumpahead()" methods have no effect and are ignored. The
   "getstate()" and "setstate()" methods raise "NotImplementedError"
   if called.

   New in version 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]

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.

  Wichmann, B. A. & Hill, I. D., “Algorithm AS 183: An efficient and
  portable pseudo-random number generator”, Applied Statistics 31
  (1982) 188-190.

  Complementary-Multiply-with-Carry recipe for a compatible
  alternative random number generator with a long period and
  comparatively simple update operations.
