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Guido van Rossume7b146f2000-02-04 15:28:42 +00001"""Random variable generators.
Guido van Rossumff03b1a1994-03-09 12:55:02 +00002
Tim Petersd7b5e882001-01-25 03:36:26 +00003 integers
4 --------
5 uniform within range
6
7 sequences
8 ---------
9 pick random element
Raymond Hettingerf24eb352002-11-12 17:41:57 +000010 pick random sample
Tim Petersd7b5e882001-01-25 03:36:26 +000011 generate random permutation
12
Guido van Rossume7b146f2000-02-04 15:28:42 +000013 distributions on the real line:
14 ------------------------------
Tim Petersd7b5e882001-01-25 03:36:26 +000015 uniform
Guido van Rossume7b146f2000-02-04 15:28:42 +000016 normal (Gaussian)
17 lognormal
18 negative exponential
19 gamma
20 beta
Raymond Hettinger40f62172002-12-29 23:03:38 +000021 pareto
22 Weibull
Guido van Rossumff03b1a1994-03-09 12:55:02 +000023
Guido van Rossume7b146f2000-02-04 15:28:42 +000024 distributions on the circle (angles 0 to 2pi)
25 ---------------------------------------------
26 circular uniform
27 von Mises
28
Raymond Hettinger40f62172002-12-29 23:03:38 +000029General notes on the underlying Mersenne Twister core generator:
Guido van Rossume7b146f2000-02-04 15:28:42 +000030
Raymond Hettinger40f62172002-12-29 23:03:38 +000031* The period is 2**19937-1.
Tim Peters0e115952006-06-10 22:51:45 +000032* It is one of the most extensively tested generators in existence.
33* Without a direct way to compute N steps forward, the semantics of
34 jumpahead(n) are weakened to simply jump to another distant state and rely
35 on the large period to avoid overlapping sequences.
36* The random() method is implemented in C, executes in a single Python step,
37 and is, therefore, threadsafe.
Tim Peterse360d952001-01-26 10:00:39 +000038
Guido van Rossume7b146f2000-02-04 15:28:42 +000039"""
Guido van Rossumd03e1191998-05-29 17:51:31 +000040
Raymond Hettinger2f726e92003-10-05 09:09:15 +000041from warnings import warn as _warn
42from types import MethodType as _MethodType, BuiltinMethodType as _BuiltinMethodType
Raymond Hettinger91e27c22005-08-19 01:36:35 +000043from math import log as _log, exp as _exp, pi as _pi, e as _e, ceil as _ceil
Tim Petersd7b5e882001-01-25 03:36:26 +000044from math import sqrt as _sqrt, acos as _acos, cos as _cos, sin as _sin
Raymond Hettingerc1c43ca2004-09-05 00:00:42 +000045from os import urandom as _urandom
46from binascii import hexlify as _hexlify
Guido van Rossumff03b1a1994-03-09 12:55:02 +000047
Raymond Hettingerf24eb352002-11-12 17:41:57 +000048__all__ = ["Random","seed","random","uniform","randint","choice","sample",
Skip Montanaro0de65802001-02-15 22:15:14 +000049 "randrange","shuffle","normalvariate","lognormvariate",
Raymond Hettingerf8a52d32003-08-05 12:23:19 +000050 "expovariate","vonmisesvariate","gammavariate",
51 "gauss","betavariate","paretovariate","weibullvariate",
Raymond Hettinger356a4592004-08-30 06:14:31 +000052 "getstate","setstate","jumpahead", "WichmannHill", "getrandbits",
Raymond Hettinger23f12412004-09-13 22:23:21 +000053 "SystemRandom"]
Tim Petersd7b5e882001-01-25 03:36:26 +000054
55NV_MAGICCONST = 4 * _exp(-0.5)/_sqrt(2.0)
Tim Petersd7b5e882001-01-25 03:36:26 +000056TWOPI = 2.0*_pi
Tim Petersd7b5e882001-01-25 03:36:26 +000057LOG4 = _log(4.0)
Tim Petersd7b5e882001-01-25 03:36:26 +000058SG_MAGICCONST = 1.0 + _log(4.5)
Raymond Hettinger2f726e92003-10-05 09:09:15 +000059BPF = 53 # Number of bits in a float
Tim Peters7c2a85b2004-08-31 02:19:55 +000060RECIP_BPF = 2**-BPF
Tim Petersd7b5e882001-01-25 03:36:26 +000061
Raymond Hettinger356a4592004-08-30 06:14:31 +000062
Tim Petersd7b5e882001-01-25 03:36:26 +000063# Translated by Guido van Rossum from C source provided by
Raymond Hettinger40f62172002-12-29 23:03:38 +000064# Adrian Baddeley. Adapted by Raymond Hettinger for use with
Raymond Hettinger3fa19d72004-08-31 01:05:15 +000065# the Mersenne Twister and os.urandom() core generators.
Tim Petersd7b5e882001-01-25 03:36:26 +000066
Raymond Hettinger145a4a02003-01-07 10:25:55 +000067import _random
Raymond Hettinger40f62172002-12-29 23:03:38 +000068
Raymond Hettinger145a4a02003-01-07 10:25:55 +000069class Random(_random.Random):
Raymond Hettingerc32f0332002-05-23 19:44:49 +000070 """Random number generator base class used by bound module functions.
71
72 Used to instantiate instances of Random to get generators that don't
73 share state. Especially useful for multi-threaded programs, creating
74 a different instance of Random for each thread, and using the jumpahead()
75 method to ensure that the generated sequences seen by each thread don't
76 overlap.
77
78 Class Random can also be subclassed if you want to use a different basic
79 generator of your own devising: in that case, override the following
80 methods: random(), seed(), getstate(), setstate() and jumpahead().
Raymond Hettinger2f726e92003-10-05 09:09:15 +000081 Optionally, implement a getrandombits() method so that randrange()
82 can cover arbitrarily large ranges.
Raymond Hettingeref4d4bd2002-05-23 23:58:17 +000083
Raymond Hettingerc32f0332002-05-23 19:44:49 +000084 """
Tim Petersd7b5e882001-01-25 03:36:26 +000085
Raymond Hettinger40f62172002-12-29 23:03:38 +000086 VERSION = 2 # used by getstate/setstate
Tim Petersd7b5e882001-01-25 03:36:26 +000087
88 def __init__(self, x=None):
89 """Initialize an instance.
90
91 Optional argument x controls seeding, as for Random.seed().
92 """
93
94 self.seed(x)
Raymond Hettinger40f62172002-12-29 23:03:38 +000095 self.gauss_next = None
Tim Petersd7b5e882001-01-25 03:36:26 +000096
Tim Peters0de88fc2001-02-01 04:59:18 +000097 def seed(self, a=None):
98 """Initialize internal state from hashable object.
Tim Petersd7b5e882001-01-25 03:36:26 +000099
Raymond Hettinger23f12412004-09-13 22:23:21 +0000100 None or no argument seeds from current time or from an operating
101 system specific randomness source if available.
Tim Peters0de88fc2001-02-01 04:59:18 +0000102
Tim Petersbcd725f2001-02-01 10:06:53 +0000103 If a is not None or an int or long, hash(a) is used instead.
Tim Petersd7b5e882001-01-25 03:36:26 +0000104 """
105
Raymond Hettinger3081d592003-08-09 18:30:57 +0000106 if a is None:
Raymond Hettingerc1c43ca2004-09-05 00:00:42 +0000107 try:
108 a = long(_hexlify(_urandom(16)), 16)
109 except NotImplementedError:
Raymond Hettinger356a4592004-08-30 06:14:31 +0000110 import time
111 a = long(time.time() * 256) # use fractional seconds
Raymond Hettinger356a4592004-08-30 06:14:31 +0000112
Raymond Hettinger145a4a02003-01-07 10:25:55 +0000113 super(Random, self).seed(a)
Tim Peters46c04e12002-05-05 20:40:00 +0000114 self.gauss_next = None
115
Tim Peterscd804102001-01-25 20:25:57 +0000116 def getstate(self):
117 """Return internal state; can be passed to setstate() later."""
Raymond Hettinger145a4a02003-01-07 10:25:55 +0000118 return self.VERSION, super(Random, self).getstate(), self.gauss_next
Tim Peterscd804102001-01-25 20:25:57 +0000119
120 def setstate(self, state):
121 """Restore internal state from object returned by getstate()."""
122 version = state[0]
Raymond Hettinger40f62172002-12-29 23:03:38 +0000123 if version == 2:
124 version, internalstate, self.gauss_next = state
Raymond Hettinger145a4a02003-01-07 10:25:55 +0000125 super(Random, self).setstate(internalstate)
Tim Peterscd804102001-01-25 20:25:57 +0000126 else:
127 raise ValueError("state with version %s passed to "
128 "Random.setstate() of version %s" %
129 (version, self.VERSION))
130
Tim Peterscd804102001-01-25 20:25:57 +0000131## ---- Methods below this point do not need to be overridden when
132## ---- subclassing for the purpose of using a different core generator.
133
134## -------------------- pickle support -------------------
135
136 def __getstate__(self): # for pickle
137 return self.getstate()
138
139 def __setstate__(self, state): # for pickle
140 self.setstate(state)
141
Raymond Hettinger5f078ff2003-06-24 20:29:04 +0000142 def __reduce__(self):
143 return self.__class__, (), self.getstate()
144
Tim Peterscd804102001-01-25 20:25:57 +0000145## -------------------- integer methods -------------------
146
Raymond Hettinger2f726e92003-10-05 09:09:15 +0000147 def randrange(self, start, stop=None, step=1, int=int, default=None,
148 maxwidth=1L<<BPF):
Tim Petersd7b5e882001-01-25 03:36:26 +0000149 """Choose a random item from range(start, stop[, step]).
150
151 This fixes the problem with randint() which includes the
152 endpoint; in Python this is usually not what you want.
Raymond Hettinger2f726e92003-10-05 09:09:15 +0000153 Do not supply the 'int', 'default', and 'maxwidth' arguments.
Tim Petersd7b5e882001-01-25 03:36:26 +0000154 """
155
156 # This code is a bit messy to make it fast for the
Tim Peters9146f272002-08-16 03:41:39 +0000157 # common case while still doing adequate error checking.
Tim Petersd7b5e882001-01-25 03:36:26 +0000158 istart = int(start)
159 if istart != start:
160 raise ValueError, "non-integer arg 1 for randrange()"
161 if stop is default:
162 if istart > 0:
Raymond Hettinger2f726e92003-10-05 09:09:15 +0000163 if istart >= maxwidth:
164 return self._randbelow(istart)
Tim Petersd7b5e882001-01-25 03:36:26 +0000165 return int(self.random() * istart)
166 raise ValueError, "empty range for randrange()"
Tim Peters9146f272002-08-16 03:41:39 +0000167
168 # stop argument supplied.
Tim Petersd7b5e882001-01-25 03:36:26 +0000169 istop = int(stop)
170 if istop != stop:
171 raise ValueError, "non-integer stop for randrange()"
Raymond Hettinger2f726e92003-10-05 09:09:15 +0000172 width = istop - istart
173 if step == 1 and width > 0:
Tim Peters76ca1d42003-06-19 03:46:46 +0000174 # Note that
Raymond Hettinger2f726e92003-10-05 09:09:15 +0000175 # int(istart + self.random()*width)
Tim Peters76ca1d42003-06-19 03:46:46 +0000176 # instead would be incorrect. For example, consider istart
177 # = -2 and istop = 0. Then the guts would be in
178 # -2.0 to 0.0 exclusive on both ends (ignoring that random()
179 # might return 0.0), and because int() truncates toward 0, the
180 # final result would be -1 or 0 (instead of -2 or -1).
Raymond Hettinger2f726e92003-10-05 09:09:15 +0000181 # istart + int(self.random()*width)
Tim Peters76ca1d42003-06-19 03:46:46 +0000182 # would also be incorrect, for a subtler reason: the RHS
183 # can return a long, and then randrange() would also return
184 # a long, but we're supposed to return an int (for backward
185 # compatibility).
Raymond Hettinger2f726e92003-10-05 09:09:15 +0000186
187 if width >= maxwidth:
Tim Peters58eb11c2004-01-18 20:29:55 +0000188 return int(istart + self._randbelow(width))
Raymond Hettinger2f726e92003-10-05 09:09:15 +0000189 return int(istart + int(self.random()*width))
Tim Petersd7b5e882001-01-25 03:36:26 +0000190 if step == 1:
Raymond Hettinger2f726e92003-10-05 09:09:15 +0000191 raise ValueError, "empty range for randrange() (%d,%d, %d)" % (istart, istop, width)
Tim Peters9146f272002-08-16 03:41:39 +0000192
193 # Non-unit step argument supplied.
Tim Petersd7b5e882001-01-25 03:36:26 +0000194 istep = int(step)
195 if istep != step:
196 raise ValueError, "non-integer step for randrange()"
197 if istep > 0:
Raymond Hettingerffdb8bb2004-09-27 15:29:05 +0000198 n = (width + istep - 1) // istep
Tim Petersd7b5e882001-01-25 03:36:26 +0000199 elif istep < 0:
Raymond Hettingerffdb8bb2004-09-27 15:29:05 +0000200 n = (width + istep + 1) // istep
Tim Petersd7b5e882001-01-25 03:36:26 +0000201 else:
202 raise ValueError, "zero step for randrange()"
203
204 if n <= 0:
205 raise ValueError, "empty range for randrange()"
Raymond Hettinger2f726e92003-10-05 09:09:15 +0000206
207 if n >= maxwidth:
Raymond Hettinger94547f72006-12-20 06:42:06 +0000208 return istart + istep*self._randbelow(n)
Tim Petersd7b5e882001-01-25 03:36:26 +0000209 return istart + istep*int(self.random() * n)
210
211 def randint(self, a, b):
Tim Peterscd804102001-01-25 20:25:57 +0000212 """Return random integer in range [a, b], including both end points.
Tim Petersd7b5e882001-01-25 03:36:26 +0000213 """
214
215 return self.randrange(a, b+1)
216
Raymond Hettinger2f726e92003-10-05 09:09:15 +0000217 def _randbelow(self, n, _log=_log, int=int, _maxwidth=1L<<BPF,
218 _Method=_MethodType, _BuiltinMethod=_BuiltinMethodType):
219 """Return a random int in the range [0,n)
220
221 Handles the case where n has more bits than returned
222 by a single call to the underlying generator.
223 """
224
225 try:
226 getrandbits = self.getrandbits
227 except AttributeError:
228 pass
229 else:
230 # Only call self.getrandbits if the original random() builtin method
231 # has not been overridden or if a new getrandbits() was supplied.
232 # This assures that the two methods correspond.
233 if type(self.random) is _BuiltinMethod or type(getrandbits) is _Method:
234 k = int(1.00001 + _log(n-1, 2.0)) # 2**k > n-1 > 2**(k-2)
235 r = getrandbits(k)
236 while r >= n:
237 r = getrandbits(k)
238 return r
239 if n >= _maxwidth:
240 _warn("Underlying random() generator does not supply \n"
241 "enough bits to choose from a population range this large")
242 return int(self.random() * n)
243
Tim Peterscd804102001-01-25 20:25:57 +0000244## -------------------- sequence methods -------------------
245
Tim Petersd7b5e882001-01-25 03:36:26 +0000246 def choice(self, seq):
247 """Choose a random element from a non-empty sequence."""
Raymond Hettinger5dae5052004-06-07 02:07:15 +0000248 return seq[int(self.random() * len(seq))] # raises IndexError if seq is empty
Tim Petersd7b5e882001-01-25 03:36:26 +0000249
250 def shuffle(self, x, random=None, int=int):
251 """x, random=random.random -> shuffle list x in place; return None.
252
253 Optional arg random is a 0-argument function returning a random
254 float in [0.0, 1.0); by default, the standard random.random.
Tim Petersd7b5e882001-01-25 03:36:26 +0000255 """
256
257 if random is None:
258 random = self.random
Raymond Hettinger85c20a42003-11-06 14:06:48 +0000259 for i in reversed(xrange(1, len(x))):
Tim Peterscd804102001-01-25 20:25:57 +0000260 # pick an element in x[:i+1] with which to exchange x[i]
Tim Petersd7b5e882001-01-25 03:36:26 +0000261 j = int(random() * (i+1))
262 x[i], x[j] = x[j], x[i]
263
Raymond Hettingerfdbe5222003-06-13 07:01:51 +0000264 def sample(self, population, k):
Raymond Hettingerf24eb352002-11-12 17:41:57 +0000265 """Chooses k unique random elements from a population sequence.
266
Raymond Hettingerc0b40342002-11-13 15:26:37 +0000267 Returns a new list containing elements from the population while
268 leaving the original population unchanged. The resulting list is
269 in selection order so that all sub-slices will also be valid random
270 samples. This allows raffle winners (the sample) to be partitioned
271 into grand prize and second place winners (the subslices).
Raymond Hettingerf24eb352002-11-12 17:41:57 +0000272
Raymond Hettingerc0b40342002-11-13 15:26:37 +0000273 Members of the population need not be hashable or unique. If the
274 population contains repeats, then each occurrence is a possible
275 selection in the sample.
Raymond Hettingerf24eb352002-11-12 17:41:57 +0000276
Raymond Hettingerc0b40342002-11-13 15:26:37 +0000277 To choose a sample in a range of integers, use xrange as an argument.
278 This is especially fast and space efficient for sampling from a
279 large population: sample(xrange(10000000), 60)
Raymond Hettingerf24eb352002-11-12 17:41:57 +0000280 """
281
Tim Petersc17976e2006-04-01 00:26:53 +0000282 # XXX Although the documentation says `population` is "a sequence",
283 # XXX attempts are made to cater to any iterable with a __len__
284 # XXX method. This has had mixed success. Examples from both
285 # XXX sides: sets work fine, and should become officially supported;
286 # XXX dicts are much harder, and have failed in various subtle
287 # XXX ways across attempts. Support for mapping types should probably
288 # XXX be dropped (and users should pass mapping.keys() or .values()
289 # XXX explicitly).
290
Raymond Hettingerc0b40342002-11-13 15:26:37 +0000291 # Sampling without replacement entails tracking either potential
Raymond Hettinger91e27c22005-08-19 01:36:35 +0000292 # selections (the pool) in a list or previous selections in a set.
Raymond Hettingerc0b40342002-11-13 15:26:37 +0000293
Jeremy Hylton2b55d352004-02-23 17:27:57 +0000294 # When the number of selections is small compared to the
295 # population, then tracking selections is efficient, requiring
Raymond Hettinger91e27c22005-08-19 01:36:35 +0000296 # only a small set and an occasional reselection. For
Jeremy Hylton2b55d352004-02-23 17:27:57 +0000297 # a larger number of selections, the pool tracking method is
298 # preferred since the list takes less space than the
Raymond Hettinger91e27c22005-08-19 01:36:35 +0000299 # set and it doesn't suffer from frequent reselections.
Raymond Hettingerc0b40342002-11-13 15:26:37 +0000300
Raymond Hettingerf24eb352002-11-12 17:41:57 +0000301 n = len(population)
302 if not 0 <= k <= n:
303 raise ValueError, "sample larger than population"
Raymond Hettinger8b9aa8d2003-01-04 05:20:33 +0000304 random = self.random
Raymond Hettingerfdbe5222003-06-13 07:01:51 +0000305 _int = int
Raymond Hettingerc0b40342002-11-13 15:26:37 +0000306 result = [None] * k
Raymond Hettinger91e27c22005-08-19 01:36:35 +0000307 setsize = 21 # size of a small set minus size of an empty list
308 if k > 5:
Tim Peters9e34c042005-08-26 15:20:46 +0000309 setsize += 4 ** _ceil(_log(k * 3, 4)) # table size for big sets
Tim Petersc17976e2006-04-01 00:26:53 +0000310 if n <= setsize or hasattr(population, "keys"):
311 # An n-length list is smaller than a k-length set, or this is a
312 # mapping type so the other algorithm wouldn't work.
Raymond Hettinger311f4192002-11-18 09:01:24 +0000313 pool = list(population)
314 for i in xrange(k): # invariant: non-selected at [0,n-i)
Raymond Hettingerfdbe5222003-06-13 07:01:51 +0000315 j = _int(random() * (n-i))
Raymond Hettinger311f4192002-11-18 09:01:24 +0000316 result[i] = pool[j]
Raymond Hettinger8b9aa8d2003-01-04 05:20:33 +0000317 pool[j] = pool[n-i-1] # move non-selected item into vacancy
Raymond Hettingerc0b40342002-11-13 15:26:37 +0000318 else:
Raymond Hettinger66d09f12003-09-06 04:25:54 +0000319 try:
Raymond Hettinger3c3346d2006-03-29 09:13:13 +0000320 selected = set()
321 selected_add = selected.add
322 for i in xrange(k):
Raymond Hettingerfdbe5222003-06-13 07:01:51 +0000323 j = _int(random() * n)
Raymond Hettinger3c3346d2006-03-29 09:13:13 +0000324 while j in selected:
325 j = _int(random() * n)
326 selected_add(j)
327 result[i] = population[j]
Tim Petersc17976e2006-04-01 00:26:53 +0000328 except (TypeError, KeyError): # handle (at least) sets
Raymond Hettinger3c3346d2006-03-29 09:13:13 +0000329 if isinstance(population, list):
330 raise
Tim Petersc17976e2006-04-01 00:26:53 +0000331 return self.sample(tuple(population), k)
Raymond Hettinger311f4192002-11-18 09:01:24 +0000332 return result
Raymond Hettingerf24eb352002-11-12 17:41:57 +0000333
Tim Peterscd804102001-01-25 20:25:57 +0000334## -------------------- real-valued distributions -------------------
335
336## -------------------- uniform distribution -------------------
Tim Petersd7b5e882001-01-25 03:36:26 +0000337
338 def uniform(self, a, b):
339 """Get a random number in the range [a, b)."""
340 return a + (b-a) * self.random()
Guido van Rossumff03b1a1994-03-09 12:55:02 +0000341
Tim Peterscd804102001-01-25 20:25:57 +0000342## -------------------- normal distribution --------------------
Guido van Rossumff03b1a1994-03-09 12:55:02 +0000343
Tim Petersd7b5e882001-01-25 03:36:26 +0000344 def normalvariate(self, mu, sigma):
Raymond Hettingerc32f0332002-05-23 19:44:49 +0000345 """Normal distribution.
346
347 mu is the mean, and sigma is the standard deviation.
Raymond Hettingeref4d4bd2002-05-23 23:58:17 +0000348
Raymond Hettingerc32f0332002-05-23 19:44:49 +0000349 """
Tim Petersd7b5e882001-01-25 03:36:26 +0000350 # mu = mean, sigma = standard deviation
Guido van Rossumff03b1a1994-03-09 12:55:02 +0000351
Tim Petersd7b5e882001-01-25 03:36:26 +0000352 # Uses Kinderman and Monahan method. Reference: Kinderman,
353 # A.J. and Monahan, J.F., "Computer generation of random
354 # variables using the ratio of uniform deviates", ACM Trans
355 # Math Software, 3, (1977), pp257-260.
Guido van Rossumff03b1a1994-03-09 12:55:02 +0000356
Tim Petersd7b5e882001-01-25 03:36:26 +0000357 random = self.random
Raymond Hettinger42406e62005-04-30 09:02:51 +0000358 while 1:
Tim Peters0c9886d2001-01-15 01:18:21 +0000359 u1 = random()
Raymond Hettinger73ced7e2003-01-04 09:26:32 +0000360 u2 = 1.0 - random()
Tim Petersd7b5e882001-01-25 03:36:26 +0000361 z = NV_MAGICCONST*(u1-0.5)/u2
362 zz = z*z/4.0
363 if zz <= -_log(u2):
364 break
365 return mu + z*sigma
Guido van Rossumff03b1a1994-03-09 12:55:02 +0000366
Tim Peterscd804102001-01-25 20:25:57 +0000367## -------------------- lognormal distribution --------------------
Tim Petersd7b5e882001-01-25 03:36:26 +0000368
369 def lognormvariate(self, mu, sigma):
Raymond Hettingerc32f0332002-05-23 19:44:49 +0000370 """Log normal distribution.
371
372 If you take the natural logarithm of this distribution, you'll get a
373 normal distribution with mean mu and standard deviation sigma.
374 mu can have any value, and sigma must be greater than zero.
Raymond Hettingeref4d4bd2002-05-23 23:58:17 +0000375
Raymond Hettingerc32f0332002-05-23 19:44:49 +0000376 """
Tim Petersd7b5e882001-01-25 03:36:26 +0000377 return _exp(self.normalvariate(mu, sigma))
378
Tim Peterscd804102001-01-25 20:25:57 +0000379## -------------------- exponential distribution --------------------
Tim Petersd7b5e882001-01-25 03:36:26 +0000380
381 def expovariate(self, lambd):
Raymond Hettingerc32f0332002-05-23 19:44:49 +0000382 """Exponential distribution.
383
384 lambd is 1.0 divided by the desired mean. (The parameter would be
385 called "lambda", but that is a reserved word in Python.) Returned
386 values range from 0 to positive infinity.
Raymond Hettingeref4d4bd2002-05-23 23:58:17 +0000387
Raymond Hettingerc32f0332002-05-23 19:44:49 +0000388 """
Tim Petersd7b5e882001-01-25 03:36:26 +0000389 # lambd: rate lambd = 1/mean
390 # ('lambda' is a Python reserved word)
391
392 random = self.random
Tim Peters0c9886d2001-01-15 01:18:21 +0000393 u = random()
394 while u <= 1e-7:
395 u = random()
Tim Petersd7b5e882001-01-25 03:36:26 +0000396 return -_log(u)/lambd
Guido van Rossumff03b1a1994-03-09 12:55:02 +0000397
Tim Peterscd804102001-01-25 20:25:57 +0000398## -------------------- von Mises distribution --------------------
Guido van Rossumff03b1a1994-03-09 12:55:02 +0000399
Tim Petersd7b5e882001-01-25 03:36:26 +0000400 def vonmisesvariate(self, mu, kappa):
Raymond Hettingerc32f0332002-05-23 19:44:49 +0000401 """Circular data distribution.
Raymond Hettingeref4d4bd2002-05-23 23:58:17 +0000402
Raymond Hettingerc32f0332002-05-23 19:44:49 +0000403 mu is the mean angle, expressed in radians between 0 and 2*pi, and
404 kappa is the concentration parameter, which must be greater than or
405 equal to zero. If kappa is equal to zero, this distribution reduces
406 to a uniform random angle over the range 0 to 2*pi.
Raymond Hettingeref4d4bd2002-05-23 23:58:17 +0000407
Raymond Hettingerc32f0332002-05-23 19:44:49 +0000408 """
Tim Petersd7b5e882001-01-25 03:36:26 +0000409 # mu: mean angle (in radians between 0 and 2*pi)
410 # kappa: concentration parameter kappa (>= 0)
411 # if kappa = 0 generate uniform random angle
412
413 # Based upon an algorithm published in: Fisher, N.I.,
414 # "Statistical Analysis of Circular Data", Cambridge
415 # University Press, 1993.
416
417 # Thanks to Magnus Kessler for a correction to the
418 # implementation of step 4.
419
420 random = self.random
421 if kappa <= 1e-6:
422 return TWOPI * random()
423
424 a = 1.0 + _sqrt(1.0 + 4.0 * kappa * kappa)
425 b = (a - _sqrt(2.0 * a))/(2.0 * kappa)
426 r = (1.0 + b * b)/(2.0 * b)
Guido van Rossumff03b1a1994-03-09 12:55:02 +0000427
Raymond Hettinger42406e62005-04-30 09:02:51 +0000428 while 1:
Tim Peters0c9886d2001-01-15 01:18:21 +0000429 u1 = random()
Tim Petersd7b5e882001-01-25 03:36:26 +0000430
431 z = _cos(_pi * u1)
432 f = (1.0 + r * z)/(r + z)
433 c = kappa * (r - f)
434
435 u2 = random()
436
Raymond Hettinger42406e62005-04-30 09:02:51 +0000437 if u2 < c * (2.0 - c) or u2 <= c * _exp(1.0 - c):
Tim Peters0c9886d2001-01-15 01:18:21 +0000438 break
Tim Petersd7b5e882001-01-25 03:36:26 +0000439
440 u3 = random()
441 if u3 > 0.5:
442 theta = (mu % TWOPI) + _acos(f)
443 else:
444 theta = (mu % TWOPI) - _acos(f)
445
446 return theta
447
Tim Peterscd804102001-01-25 20:25:57 +0000448## -------------------- gamma distribution --------------------
Tim Petersd7b5e882001-01-25 03:36:26 +0000449
450 def gammavariate(self, alpha, beta):
Raymond Hettingerc32f0332002-05-23 19:44:49 +0000451 """Gamma distribution. Not the gamma function!
452
453 Conditions on the parameters are alpha > 0 and beta > 0.
454
455 """
Tim Peters8ac14952002-05-23 15:15:30 +0000456
Raymond Hettingerb760efb2002-05-14 06:40:34 +0000457 # alpha > 0, beta > 0, mean is alpha*beta, variance is alpha*beta**2
Tim Peters8ac14952002-05-23 15:15:30 +0000458
Guido van Rossum570764d2002-05-14 14:08:12 +0000459 # Warning: a few older sources define the gamma distribution in terms
460 # of alpha > -1.0
461 if alpha <= 0.0 or beta <= 0.0:
462 raise ValueError, 'gammavariate: alpha and beta must be > 0.0'
Tim Peters8ac14952002-05-23 15:15:30 +0000463
Tim Petersd7b5e882001-01-25 03:36:26 +0000464 random = self.random
Tim Petersd7b5e882001-01-25 03:36:26 +0000465 if alpha > 1.0:
466
467 # Uses R.C.H. Cheng, "The generation of Gamma
468 # variables with non-integral shape parameters",
469 # Applied Statistics, (1977), 26, No. 1, p71-74
470
Raymond Hettingerca6cdc22002-05-13 23:40:14 +0000471 ainv = _sqrt(2.0 * alpha - 1.0)
472 bbb = alpha - LOG4
473 ccc = alpha + ainv
Tim Peters8ac14952002-05-23 15:15:30 +0000474
Raymond Hettinger42406e62005-04-30 09:02:51 +0000475 while 1:
Tim Petersd7b5e882001-01-25 03:36:26 +0000476 u1 = random()
Raymond Hettinger73ced7e2003-01-04 09:26:32 +0000477 if not 1e-7 < u1 < .9999999:
478 continue
479 u2 = 1.0 - random()
Tim Petersd7b5e882001-01-25 03:36:26 +0000480 v = _log(u1/(1.0-u1))/ainv
481 x = alpha*_exp(v)
482 z = u1*u1*u2
483 r = bbb+ccc*v-x
484 if r + SG_MAGICCONST - 4.5*z >= 0.0 or r >= _log(z):
Raymond Hettingerb760efb2002-05-14 06:40:34 +0000485 return x * beta
Tim Petersd7b5e882001-01-25 03:36:26 +0000486
487 elif alpha == 1.0:
488 # expovariate(1)
489 u = random()
490 while u <= 1e-7:
491 u = random()
Raymond Hettingerb760efb2002-05-14 06:40:34 +0000492 return -_log(u) * beta
Tim Petersd7b5e882001-01-25 03:36:26 +0000493
494 else: # alpha is between 0 and 1 (exclusive)
495
496 # Uses ALGORITHM GS of Statistical Computing - Kennedy & Gentle
497
Raymond Hettinger42406e62005-04-30 09:02:51 +0000498 while 1:
Tim Petersd7b5e882001-01-25 03:36:26 +0000499 u = random()
500 b = (_e + alpha)/_e
501 p = b*u
502 if p <= 1.0:
Raymond Hettinger42406e62005-04-30 09:02:51 +0000503 x = p ** (1.0/alpha)
Tim Petersd7b5e882001-01-25 03:36:26 +0000504 else:
Tim Petersd7b5e882001-01-25 03:36:26 +0000505 x = -_log((b-p)/alpha)
506 u1 = random()
Raymond Hettinger42406e62005-04-30 09:02:51 +0000507 if p > 1.0:
508 if u1 <= x ** (alpha - 1.0):
509 break
510 elif u1 <= _exp(-x):
Tim Petersd7b5e882001-01-25 03:36:26 +0000511 break
Raymond Hettingerb760efb2002-05-14 06:40:34 +0000512 return x * beta
513
Tim Peterscd804102001-01-25 20:25:57 +0000514## -------------------- Gauss (faster alternative) --------------------
Guido van Rossum95bfcda1994-03-09 14:21:05 +0000515
Tim Petersd7b5e882001-01-25 03:36:26 +0000516 def gauss(self, mu, sigma):
Raymond Hettingerc32f0332002-05-23 19:44:49 +0000517 """Gaussian distribution.
518
519 mu is the mean, and sigma is the standard deviation. This is
520 slightly faster than the normalvariate() function.
521
522 Not thread-safe without a lock around calls.
Raymond Hettingeref4d4bd2002-05-23 23:58:17 +0000523
Raymond Hettingerc32f0332002-05-23 19:44:49 +0000524 """
Guido van Rossumcc32ac91994-03-15 16:10:24 +0000525
Tim Petersd7b5e882001-01-25 03:36:26 +0000526 # When x and y are two variables from [0, 1), uniformly
527 # distributed, then
528 #
529 # cos(2*pi*x)*sqrt(-2*log(1-y))
530 # sin(2*pi*x)*sqrt(-2*log(1-y))
531 #
532 # are two *independent* variables with normal distribution
533 # (mu = 0, sigma = 1).
534 # (Lambert Meertens)
535 # (corrected version; bug discovered by Mike Miller, fixed by LM)
Guido van Rossumcc32ac91994-03-15 16:10:24 +0000536
Tim Petersd7b5e882001-01-25 03:36:26 +0000537 # Multithreading note: When two threads call this function
538 # simultaneously, it is possible that they will receive the
539 # same return value. The window is very small though. To
540 # avoid this, you have to use a lock around all calls. (I
541 # didn't want to slow this down in the serial case by using a
542 # lock here.)
Guido van Rossumd03e1191998-05-29 17:51:31 +0000543
Tim Petersd7b5e882001-01-25 03:36:26 +0000544 random = self.random
545 z = self.gauss_next
546 self.gauss_next = None
547 if z is None:
548 x2pi = random() * TWOPI
549 g2rad = _sqrt(-2.0 * _log(1.0 - random()))
550 z = _cos(x2pi) * g2rad
551 self.gauss_next = _sin(x2pi) * g2rad
Guido van Rossumcc32ac91994-03-15 16:10:24 +0000552
Tim Petersd7b5e882001-01-25 03:36:26 +0000553 return mu + z*sigma
Guido van Rossum95bfcda1994-03-09 14:21:05 +0000554
Tim Peterscd804102001-01-25 20:25:57 +0000555## -------------------- beta --------------------
Tim Peters85e2e472001-01-26 06:49:56 +0000556## See
557## http://sourceforge.net/bugs/?func=detailbug&bug_id=130030&group_id=5470
558## for Ivan Frohne's insightful analysis of why the original implementation:
559##
560## def betavariate(self, alpha, beta):
561## # Discrete Event Simulation in C, pp 87-88.
562##
563## y = self.expovariate(alpha)
564## z = self.expovariate(1.0/beta)
565## return z/(y+z)
566##
567## was dead wrong, and how it probably got that way.
Guido van Rossum95bfcda1994-03-09 14:21:05 +0000568
Tim Petersd7b5e882001-01-25 03:36:26 +0000569 def betavariate(self, alpha, beta):
Raymond Hettingerc32f0332002-05-23 19:44:49 +0000570 """Beta distribution.
571
Raymond Hettinger1b0ce852007-01-19 18:07:18 +0000572 Conditions on the parameters are alpha > 0 and beta > 0.
Raymond Hettingerc32f0332002-05-23 19:44:49 +0000573 Returned values range between 0 and 1.
Raymond Hettingeref4d4bd2002-05-23 23:58:17 +0000574
Raymond Hettingerc32f0332002-05-23 19:44:49 +0000575 """
Raymond Hettingeref4d4bd2002-05-23 23:58:17 +0000576
Tim Peters85e2e472001-01-26 06:49:56 +0000577 # This version due to Janne Sinkkonen, and matches all the std
578 # texts (e.g., Knuth Vol 2 Ed 3 pg 134 "the beta distribution").
579 y = self.gammavariate(alpha, 1.)
580 if y == 0:
581 return 0.0
582 else:
583 return y / (y + self.gammavariate(beta, 1.))
Guido van Rossum95bfcda1994-03-09 14:21:05 +0000584
Tim Peterscd804102001-01-25 20:25:57 +0000585## -------------------- Pareto --------------------
Guido van Rossumcf4559a1997-12-02 02:47:39 +0000586
Tim Petersd7b5e882001-01-25 03:36:26 +0000587 def paretovariate(self, alpha):
Raymond Hettingerc32f0332002-05-23 19:44:49 +0000588 """Pareto distribution. alpha is the shape parameter."""
Tim Petersd7b5e882001-01-25 03:36:26 +0000589 # Jain, pg. 495
Guido van Rossumcf4559a1997-12-02 02:47:39 +0000590
Raymond Hettinger73ced7e2003-01-04 09:26:32 +0000591 u = 1.0 - self.random()
Tim Petersd7b5e882001-01-25 03:36:26 +0000592 return 1.0 / pow(u, 1.0/alpha)
Guido van Rossumcf4559a1997-12-02 02:47:39 +0000593
Tim Peterscd804102001-01-25 20:25:57 +0000594## -------------------- Weibull --------------------
Guido van Rossumcf4559a1997-12-02 02:47:39 +0000595
Tim Petersd7b5e882001-01-25 03:36:26 +0000596 def weibullvariate(self, alpha, beta):
Raymond Hettingerc32f0332002-05-23 19:44:49 +0000597 """Weibull distribution.
598
599 alpha is the scale parameter and beta is the shape parameter.
Raymond Hettingeref4d4bd2002-05-23 23:58:17 +0000600
Raymond Hettingerc32f0332002-05-23 19:44:49 +0000601 """
Tim Petersd7b5e882001-01-25 03:36:26 +0000602 # Jain, pg. 499; bug fix courtesy Bill Arms
Guido van Rossumcf4559a1997-12-02 02:47:39 +0000603
Raymond Hettinger73ced7e2003-01-04 09:26:32 +0000604 u = 1.0 - self.random()
Tim Petersd7b5e882001-01-25 03:36:26 +0000605 return alpha * pow(-_log(u), 1.0/beta)
Guido van Rossum6c395ba1999-08-18 13:53:28 +0000606
Raymond Hettinger40f62172002-12-29 23:03:38 +0000607## -------------------- Wichmann-Hill -------------------
608
609class WichmannHill(Random):
610
611 VERSION = 1 # used by getstate/setstate
612
613 def seed(self, a=None):
614 """Initialize internal state from hashable object.
615
Raymond Hettinger23f12412004-09-13 22:23:21 +0000616 None or no argument seeds from current time or from an operating
617 system specific randomness source if available.
Raymond Hettinger40f62172002-12-29 23:03:38 +0000618
619 If a is not None or an int or long, hash(a) is used instead.
620
621 If a is an int or long, a is used directly. Distinct values between
622 0 and 27814431486575L inclusive are guaranteed to yield distinct
623 internal states (this guarantee is specific to the default
624 Wichmann-Hill generator).
625 """
626
627 if a is None:
Raymond Hettingerc1c43ca2004-09-05 00:00:42 +0000628 try:
629 a = long(_hexlify(_urandom(16)), 16)
630 except NotImplementedError:
Raymond Hettinger356a4592004-08-30 06:14:31 +0000631 import time
632 a = long(time.time() * 256) # use fractional seconds
Raymond Hettinger40f62172002-12-29 23:03:38 +0000633
634 if not isinstance(a, (int, long)):
635 a = hash(a)
636
637 a, x = divmod(a, 30268)
638 a, y = divmod(a, 30306)
639 a, z = divmod(a, 30322)
640 self._seed = int(x)+1, int(y)+1, int(z)+1
641
642 self.gauss_next = None
643
644 def random(self):
645 """Get the next random number in the range [0.0, 1.0)."""
646
647 # Wichman-Hill random number generator.
648 #
649 # Wichmann, B. A. & Hill, I. D. (1982)
650 # Algorithm AS 183:
651 # An efficient and portable pseudo-random number generator
652 # Applied Statistics 31 (1982) 188-190
653 #
654 # see also:
655 # Correction to Algorithm AS 183
656 # Applied Statistics 33 (1984) 123
657 #
658 # McLeod, A. I. (1985)
659 # A remark on Algorithm AS 183
660 # Applied Statistics 34 (1985),198-200
661
662 # This part is thread-unsafe:
663 # BEGIN CRITICAL SECTION
664 x, y, z = self._seed
665 x = (171 * x) % 30269
666 y = (172 * y) % 30307
667 z = (170 * z) % 30323
668 self._seed = x, y, z
669 # END CRITICAL SECTION
670
671 # Note: on a platform using IEEE-754 double arithmetic, this can
672 # never return 0.0 (asserted by Tim; proof too long for a comment).
673 return (x/30269.0 + y/30307.0 + z/30323.0) % 1.0
674
675 def getstate(self):
676 """Return internal state; can be passed to setstate() later."""
677 return self.VERSION, self._seed, self.gauss_next
678
679 def setstate(self, state):
680 """Restore internal state from object returned by getstate()."""
681 version = state[0]
682 if version == 1:
683 version, self._seed, self.gauss_next = state
684 else:
685 raise ValueError("state with version %s passed to "
686 "Random.setstate() of version %s" %
687 (version, self.VERSION))
688
689 def jumpahead(self, n):
690 """Act as if n calls to random() were made, but quickly.
691
692 n is an int, greater than or equal to 0.
693
694 Example use: If you have 2 threads and know that each will
695 consume no more than a million random numbers, create two Random
696 objects r1 and r2, then do
697 r2.setstate(r1.getstate())
698 r2.jumpahead(1000000)
699 Then r1 and r2 will use guaranteed-disjoint segments of the full
700 period.
701 """
702
703 if not n >= 0:
704 raise ValueError("n must be >= 0")
705 x, y, z = self._seed
706 x = int(x * pow(171, n, 30269)) % 30269
707 y = int(y * pow(172, n, 30307)) % 30307
708 z = int(z * pow(170, n, 30323)) % 30323
709 self._seed = x, y, z
710
711 def __whseed(self, x=0, y=0, z=0):
712 """Set the Wichmann-Hill seed from (x, y, z).
713
714 These must be integers in the range [0, 256).
715 """
716
717 if not type(x) == type(y) == type(z) == int:
718 raise TypeError('seeds must be integers')
719 if not (0 <= x < 256 and 0 <= y < 256 and 0 <= z < 256):
720 raise ValueError('seeds must be in range(0, 256)')
721 if 0 == x == y == z:
722 # Initialize from current time
723 import time
724 t = long(time.time() * 256)
725 t = int((t&0xffffff) ^ (t>>24))
726 t, x = divmod(t, 256)
727 t, y = divmod(t, 256)
728 t, z = divmod(t, 256)
729 # Zero is a poor seed, so substitute 1
730 self._seed = (x or 1, y or 1, z or 1)
731
732 self.gauss_next = None
733
734 def whseed(self, a=None):
735 """Seed from hashable object's hash code.
736
737 None or no argument seeds from current time. It is not guaranteed
738 that objects with distinct hash codes lead to distinct internal
739 states.
740
741 This is obsolete, provided for compatibility with the seed routine
742 used prior to Python 2.1. Use the .seed() method instead.
743 """
744
745 if a is None:
746 self.__whseed()
747 return
748 a = hash(a)
749 a, x = divmod(a, 256)
750 a, y = divmod(a, 256)
751 a, z = divmod(a, 256)
752 x = (x + a) % 256 or 1
753 y = (y + a) % 256 or 1
754 z = (z + a) % 256 or 1
755 self.__whseed(x, y, z)
756
Raymond Hettinger23f12412004-09-13 22:23:21 +0000757## --------------- Operating System Random Source ------------------
Raymond Hettinger356a4592004-08-30 06:14:31 +0000758
Raymond Hettinger23f12412004-09-13 22:23:21 +0000759class SystemRandom(Random):
760 """Alternate random number generator using sources provided
761 by the operating system (such as /dev/urandom on Unix or
762 CryptGenRandom on Windows).
Raymond Hettinger356a4592004-08-30 06:14:31 +0000763
764 Not available on all systems (see os.urandom() for details).
765 """
766
767 def random(self):
768 """Get the next random number in the range [0.0, 1.0)."""
Tim Peters7c2a85b2004-08-31 02:19:55 +0000769 return (long(_hexlify(_urandom(7)), 16) >> 3) * RECIP_BPF
Raymond Hettinger356a4592004-08-30 06:14:31 +0000770
771 def getrandbits(self, k):
772 """getrandbits(k) -> x. Generates a long int with k random bits."""
Raymond Hettinger356a4592004-08-30 06:14:31 +0000773 if k <= 0:
774 raise ValueError('number of bits must be greater than zero')
775 if k != int(k):
776 raise TypeError('number of bits should be an integer')
777 bytes = (k + 7) // 8 # bits / 8 and rounded up
778 x = long(_hexlify(_urandom(bytes)), 16)
779 return x >> (bytes * 8 - k) # trim excess bits
780
781 def _stub(self, *args, **kwds):
Raymond Hettinger23f12412004-09-13 22:23:21 +0000782 "Stub method. Not used for a system random number generator."
Raymond Hettinger356a4592004-08-30 06:14:31 +0000783 return None
784 seed = jumpahead = _stub
785
786 def _notimplemented(self, *args, **kwds):
Raymond Hettinger23f12412004-09-13 22:23:21 +0000787 "Method should not be called for a system random number generator."
788 raise NotImplementedError('System entropy source does not have state.')
Raymond Hettinger356a4592004-08-30 06:14:31 +0000789 getstate = setstate = _notimplemented
790
Tim Peterscd804102001-01-25 20:25:57 +0000791## -------------------- test program --------------------
Guido van Rossumff03b1a1994-03-09 12:55:02 +0000792
Raymond Hettinger62297132003-08-30 01:24:19 +0000793def _test_generator(n, func, args):
Tim Peters0c9886d2001-01-15 01:18:21 +0000794 import time
Raymond Hettinger62297132003-08-30 01:24:19 +0000795 print n, 'times', func.__name__
Raymond Hettingerb98154e2003-05-24 17:26:02 +0000796 total = 0.0
Tim Peters0c9886d2001-01-15 01:18:21 +0000797 sqsum = 0.0
798 smallest = 1e10
799 largest = -1e10
800 t0 = time.time()
801 for i in range(n):
Raymond Hettinger62297132003-08-30 01:24:19 +0000802 x = func(*args)
Raymond Hettingerb98154e2003-05-24 17:26:02 +0000803 total += x
Tim Peters0c9886d2001-01-15 01:18:21 +0000804 sqsum = sqsum + x*x
805 smallest = min(x, smallest)
806 largest = max(x, largest)
807 t1 = time.time()
808 print round(t1-t0, 3), 'sec,',
Raymond Hettingerb98154e2003-05-24 17:26:02 +0000809 avg = total/n
Tim Petersd7b5e882001-01-25 03:36:26 +0000810 stddev = _sqrt(sqsum/n - avg*avg)
Tim Peters0c9886d2001-01-15 01:18:21 +0000811 print 'avg %g, stddev %g, min %g, max %g' % \
812 (avg, stddev, smallest, largest)
Guido van Rossumff03b1a1994-03-09 12:55:02 +0000813
Raymond Hettingerf24eb352002-11-12 17:41:57 +0000814
815def _test(N=2000):
Raymond Hettinger62297132003-08-30 01:24:19 +0000816 _test_generator(N, random, ())
817 _test_generator(N, normalvariate, (0.0, 1.0))
818 _test_generator(N, lognormvariate, (0.0, 1.0))
819 _test_generator(N, vonmisesvariate, (0.0, 1.0))
820 _test_generator(N, gammavariate, (0.01, 1.0))
821 _test_generator(N, gammavariate, (0.1, 1.0))
822 _test_generator(N, gammavariate, (0.1, 2.0))
823 _test_generator(N, gammavariate, (0.5, 1.0))
824 _test_generator(N, gammavariate, (0.9, 1.0))
825 _test_generator(N, gammavariate, (1.0, 1.0))
826 _test_generator(N, gammavariate, (2.0, 1.0))
827 _test_generator(N, gammavariate, (20.0, 1.0))
828 _test_generator(N, gammavariate, (200.0, 1.0))
829 _test_generator(N, gauss, (0.0, 1.0))
830 _test_generator(N, betavariate, (3.0, 3.0))
Tim Peterscd804102001-01-25 20:25:57 +0000831
Tim Peters715c4c42001-01-26 22:56:56 +0000832# Create one instance, seeded from current time, and export its methods
Raymond Hettinger40f62172002-12-29 23:03:38 +0000833# as module-level functions. The functions share state across all uses
834#(both in the user's code and in the Python libraries), but that's fine
835# for most programs and is easier for the casual user than making them
836# instantiate their own Random() instance.
837
Tim Petersd7b5e882001-01-25 03:36:26 +0000838_inst = Random()
839seed = _inst.seed
840random = _inst.random
841uniform = _inst.uniform
842randint = _inst.randint
843choice = _inst.choice
844randrange = _inst.randrange
Raymond Hettingerf24eb352002-11-12 17:41:57 +0000845sample = _inst.sample
Tim Petersd7b5e882001-01-25 03:36:26 +0000846shuffle = _inst.shuffle
847normalvariate = _inst.normalvariate
848lognormvariate = _inst.lognormvariate
Tim Petersd7b5e882001-01-25 03:36:26 +0000849expovariate = _inst.expovariate
850vonmisesvariate = _inst.vonmisesvariate
851gammavariate = _inst.gammavariate
Tim Petersd7b5e882001-01-25 03:36:26 +0000852gauss = _inst.gauss
853betavariate = _inst.betavariate
854paretovariate = _inst.paretovariate
855weibullvariate = _inst.weibullvariate
856getstate = _inst.getstate
857setstate = _inst.setstate
Tim Petersd52269b2001-01-25 06:23:18 +0000858jumpahead = _inst.jumpahead
Raymond Hettinger2f726e92003-10-05 09:09:15 +0000859getrandbits = _inst.getrandbits
Tim Petersd7b5e882001-01-25 03:36:26 +0000860
Guido van Rossumff03b1a1994-03-09 12:55:02 +0000861if __name__ == '__main__':
Tim Petersd7b5e882001-01-25 03:36:26 +0000862 _test()