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, and a function to
generate a random permutation of a list in-place.
On the real line, there are functions to compute uniform, normal (Gaussian),
lognormal, negative exponential, gamma, and beta distributions.
For generating distribution of angles, the circular uniform and
von Mises distributions are 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 standard Wichmann-Hill
generator, combining three pure multiplicative congruential
generators of modulus 30269, 30307 and 30323. Its period (how many
numbers it generates before repeating the sequence exactly) is
6,953,607,871,644. While of much higher quality than the rand()
function supplied by most C libraries, the theoretical properties
are much the same as for a single linear congruential generator of
large modulus. It is not suitable for all purposes, and is completely
unsuitable for cryptographic purposes.

The functions in this module are not threadsafe: if you want to call these
functions from multiple threads, you should explicitly serialize the calls.
Else, because no critical sections are implemented internally, calls
from different threads may see the same return values.

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 (see example
below).

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.

Here's one way to create threadsafe distinct and non-overlapping generators:

def create_generators(num, delta, firstseed=None):
"""Return list of num distinct generators.
Each generator has its own unique segment of delta elements
from Random.random()'s full period.
Seed the first generator with optional arg firstseed (default
is None, to seed from current time).
"""
from random import Random
g = Random(firstseed)
result = [g]
for i in range(num - 1):
laststate = g.getstate()
g = Random()
g.setstate(laststate)
g.jumpahead(delta)
result.append(g)
return result
gens = create_generators(10, 1000000)

That creates 10 distinct generators, which can be passed out to 10
distinct threads. The generators don't share state so can be called
safely in parallel. So long as no thread calls its g.random()
more than a million times (the second argument to
create_generators(), the sequences seen by each thread will
not overlap. The period of the underlying Wichmann-Hill generator
limits how far this technique can be pushed.

Just for fun, note that since we know the period, jumpahead()
can also be used to ``move backward in time:''

>>> g = Random(42) # arbitrary
>>> g.random()
0.25420336316883324
>>> g.jumpahead(6953607871644L - 1) # move *back* one
>>> g.random()
0.25420336316883324

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(x) is used instead.
If x is an int or long, x is used directly.
Distinct values between 0 and 27814431486575L inclusive are guaranteed
to yield distinct internal states (this guarantee is specific to the
default Wichmann-Hill generator, and may not apply to subclasses
supplying their own basic generator).

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.

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 what it would be if random()
were called n times, but do so quickly. n is a
non-negative integer. 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 as
far apart as you like (up to the period of the generator).
New in version 2.1.

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.

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.

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.

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
mean - arc/2 and mean +
arc/2.

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.

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.

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.