This module implements pseudorandom 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 inplace, 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 semiopen range [0.0, 1.0). Python uses the Mersenne Twister as
the core generator. It produces 53bit precision floats and has a
period of 2**199371. 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 multithreaded
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 getrandombits()
method  this allows randrange() to produce selections
over an arbitrarily large range.
New in version 2.4:
the getrandombits() 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
WichmannHill algorithm as the core generator. Note that this WichmannHill
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:
Substituted MersenneTwister for WichmannHill.
Bookkeeping functions:

Initialize the basic random number generator.
Optional argument x can be any hashable object.
If x is omitted or
None , current system time is used;
current system time is also used to initialize the generator when the
module is first imported. 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).
Changed in version 2.4:
formerly,
operating system resources were not used.
If x is not None or an int or long,
hash(x) is used instead.
If x is an int or long, x is used directly.

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.

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
setstate() was called.
New in version 2.1.

Change the internal state to one different from and likely far away from
the current state. n is a nonnegative integer which is used to
scramble the current state vector. This is most useful in multithreaded
programs, in conjuction 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.

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:
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.

Return a random integer N such that
a <= N <= b .
Functions for sequences:

Return a random element from the nonempty sequence seq.
If seq is empty, raises IndexError.

Shuffle the sequence x in place.
The optional argument random is a 0argument 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.

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 subslices 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 realvalued 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.

Return the next random floating point number in the range [0.0, 1.0).

Return a random real number N such that
a <= N < b .
betavariate( 
alpha, beta) 

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

Exponential distribution. lambd is 1.0 divided by the desired
mean. (The parameter would be called ``lambda'', but that is a
reserved word in Python.) Returned values range from 0 to
positive infinity.
gammavariate( 
alpha, beta) 

Gamma distribution. (Not the gamma function!) Conditions on
the parameters are
alpha > 0 and beta > 0 .

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

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

Pareto distribution. alpha is the shape parameter.
weibullvariate( 
alpha, beta) 

Weibull distribution. alpha is the scale parameter and
beta is the shape parameter.
Alternative Generators:
class WichmannHill( 
[seed]) 

Class that implements the WichmannHill 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.

This is obsolete, supplied for bitlevel 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 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
623dimensionally equidistributed uniform pseudorandom
number generator'',
ACM Transactions on Modeling and Computer Simulation
Vol. 8, No. 1, January pp.330 1998.
Wichmann, B. A. & Hill, I. D., ``Algorithm AS 183:
An efficient and portable pseudorandom number generator'',
Applied Statistics 31 (1982) 188190.
 http://www.npl.co.uk/ssfm/download/abstracts.html#196
 A modern
variation of the WichmannHill generator that greatly increases
the period, and passes nowstandard statistical tests that the
original generator failed.
Release 2.5.2, documentation updated on 21st February, 2008.
See About this document... for information on suggesting changes.
