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 ensure 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.

As an example of subclassing, the `random` module provides
the `WichmannHill` class which 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.
Changed in version 2.3:
Substituted MersenneTwister for Wichmann-Hill.

Bookkeeping functions:

(`seed``[``x`]`)`-
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`x`is not`None`

or an int or long,`hash(`

is used instead. If`x`)`x`is an int or long,`x`is used directly.

(`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.

(`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`setstate()`was called. New in version 2.1.

(`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 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(`jumps to another state likely to be separated by many steps..`n`)

Functions for integers:

(`randrange``[``start,`]`stop`[`, step`]`)`-
Return a randomly selected element from
`range(`

. This is equivalent to`start`,`stop`,`step`)`choice(range(`

, but doesn't actually build a range object. New in version 1.5.2.`start`,`stop`,`step`))

(`randint``a, b`)-
Return a random integer
`N`such that

.`a`<=`N`<=`b`

Functions for sequences:

(`choice``seq`)-
Return a random element from the non-empty sequence
`seq`.

(`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(`

, the total number of permutations of`x`)`x`is larger than the period of most random number generators; this implies that most permutations of a long sequence can never be generated.

(`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

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

(`uniform``a, b`)-
Return a random real number
`N`such that

.`a`<=`N`<`b`

(`betavariate``alpha, beta`)-
Beta distribution. Conditions on the parameters are

and`alpha`> -1

. Returned values range between 0 and 1.`beta`> -1

(`cunifvariate``mean, arc`)-
Circular uniform distribution.
`mean`is the mean angle, and`arc`is the range of the distribution, centered around the mean angle. Both values must be expressed in radians, and can range between 0 and*pi*. Returned values range between

and`mean`-`arc`/2

and are normalized to between 0 and`mean`+`arc`/2*pi*.**Deprecated since release 2.3.**Instead, use`(`

.`mean`+`arc`* (random.random() - 0.5)) % math.pi

(`expovariate``lambd`)-
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

and`alpha`> 0

.`beta`> 0

(`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.

(`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*.

(`paretovariate``alpha`)-
Pareto distribution.
`alpha`is the shape parameter.

(`weibullvariate``alpha, beta`)-
Weibull distribution.
`alpha`is the scale parameter and`beta`is the shape parameter.

Alternative Generator

**class**(`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.

(`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.

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.