"""Random variable generators.     integers    --------           uniform within range     sequences    ---------           pick random element           pick random sample           generate random permutation     distributions on the real line:    ------------------------------           uniform           normal (Gaussian)           lognormal           negative exponential           gamma           beta           pareto           Weibull     distributions on the circle (angles 0 to 2pi)    ---------------------------------------------           circular uniform           von Mises General notes on the underlying Mersenne Twister core generator: * The period is 2**19937-1.* It is one of the most extensively tested generators in existence.* Without a direct way to compute N steps forward, the semantics of  jumpahead(n) are weakened to simply jump to another distant state and rely  on the large period to avoid overlapping sequences.* The random() method is implemented in C, executes in a single Python step,  and is, therefore, threadsafe. """ from warnings import warn as _warnfrom types import MethodType as _MethodType, BuiltinMethodType as _BuiltinMethodTypefrom math import log as _log, exp as _exp, pi as _pi, e as _e, ceil as _ceilfrom math import sqrt as _sqrt, acos as _acos, cos as _cos, sin as _sinfrom os import urandom as _urandomfrom binascii import hexlify as _hexlify __all__ = ["Random","seed","random","uniform","randint","choice","sample",           "randrange","shuffle","normalvariate","lognormvariate",           "expovariate","vonmisesvariate","gammavariate",           "gauss","betavariate","paretovariate","weibullvariate",           "getstate","setstate","jumpahead", "WichmannHill", "getrandbits",           "SystemRandom"] NV_MAGICCONST = 4 * _exp(-0.5)/_sqrt(2.0)TWOPI = 2.0*_piLOG4 = _log(4.0)SG_MAGICCONST = 1.0 + _log(4.5)BPF = 53        # Number of bits in a floatRECIP_BPF = 2**-BPF  # Translated by Guido van Rossum from C source provided by# Adrian Baddeley.  Adapted by Raymond Hettinger for use with# the Mersenne Twister  and os.urandom() core generators. import _random class Random(_random.Random):    """Random number generator base class used by bound module functions.     Used to instantiate instances of Random to get generators that don't    share state.  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 following    methods:  random(), seed(), getstate(), setstate() and jumpahead().    Optionally, implement a getrandombits() method so that randrange()    can cover arbitrarily large ranges.     """     VERSION = 2     # used by getstate/setstate     def __init__(self, x=None):        """Initialize an instance.         Optional argument x controls seeding, as for Random.seed().        """         self.seed(x)        self.gauss_next = None     def seed(self, a=None):        """Initialize internal state from hashable object.         None or no argument seeds from current time or from an operating        system specific randomness source if available.         If a is not None or an int or long, hash(a) is used instead.        """         if a is None:            try:                a = long(_hexlify(_urandom(16)), 16)            except NotImplementedError:                import time                a = long(time.time() * 256) # use fractional seconds         super(Random, self).seed(a)        self.gauss_next = None     def getstate(self):        """Return internal state; can be passed to setstate() later."""        return self.VERSION, super(Random, self).getstate(), self.gauss_next     def setstate(self, state):        """Restore internal state from object returned by getstate()."""        version = state[0]        if version == 2:            version, internalstate, self.gauss_next = state            super(Random, self).setstate(internalstate)        else:            raise ValueError("state with version %s passed to "                             "Random.setstate() of version %s" %                             (version, self.VERSION)) ## ---- Methods below this point do not need to be overridden when## ---- subclassing for the purpose of using a different core generator. ## -------------------- pickle support  -------------------     def __getstate__(self): # for pickle        return self.getstate()     def __setstate__(self, state):  # for pickle        self.setstate(state)     def __reduce__(self):        return self.__class__, (), self.getstate() ## -------------------- integer methods  -------------------     def randrange(self, start, stop=None, step=1, int=int, default=None,                  maxwidth=1L<<BPF):        """Choose a random item from range(start, stop[, step]).         This fixes the problem with randint() which includes the        endpoint; in Python this is usually not what you want.        Do not supply the 'int', 'default', and 'maxwidth' arguments.        """         # This code is a bit messy to make it fast for the        # common case while still doing adequate error checking.        istart = int(start)        if istart != start:            raise ValueError, "non-integer arg 1 for randrange()"        if stop is default:            if istart > 0:                if istart >= maxwidth:                    return self._randbelow(istart)                return int(self.random() * istart)            raise ValueError, "empty range for randrange()"         # stop argument supplied.        istop = int(stop)        if istop != stop:            raise ValueError, "non-integer stop for randrange()"        width = istop - istart        if step == 1 and width > 0:            # Note that            #     int(istart + self.random()*width)            # instead would be incorrect.  For example, consider istart            # = -2 and istop = 0.  Then the guts would be in            # -2.0 to 0.0 exclusive on both ends (ignoring that random()            # might return 0.0), and because int() truncates toward 0, the            # final result would be -1 or 0 (instead of -2 or -1).            #     istart + int(self.random()*width)            # would also be incorrect, for a subtler reason:  the RHS            # can return a long, and then randrange() would also return            # a long, but we're supposed to return an int (for backward            # compatibility).             if width >= maxwidth:                return int(istart + self._randbelow(width))            return int(istart + int(self.random()*width))        if step == 1:            raise ValueError, "empty range for randrange() (%d,%d, %d)" % (istart, istop, width)         # Non-unit step argument supplied.        istep = int(step)        if istep != step:            raise ValueError, "non-integer step for randrange()"        if istep > 0:            n = (width + istep - 1) // istep        elif istep < 0:            n = (width + istep + 1) // istep        else:            raise ValueError, "zero step for randrange()"         if n <= 0:            raise ValueError, "empty range for randrange()"         if n >= maxwidth:            return istart + istep*self._randbelow(n)        return istart + istep*int(self.random() * n)     def randint(self, a, b):        """Return random integer in range [a, b], including both end points.        """         return self.randrange(a, b+1)     def _randbelow(self, n, _log=_log, int=int, _maxwidth=1L<<BPF,                   _Method=_MethodType, _BuiltinMethod=_BuiltinMethodType):        """Return a random int in the range [0,n)         Handles the case where n has more bits than returned        by a single call to the underlying generator.        """         try:            getrandbits = self.getrandbits        except AttributeError:            pass        else:            # Only call self.getrandbits if the original random() builtin method            # has not been overridden or if a new getrandbits() was supplied.            # This assures that the two methods correspond.            if type(self.random) is _BuiltinMethod or type(getrandbits) is _Method:                k = int(1.00001 + _log(n-1, 2.0))   # 2**k > n-1 > 2**(k-2)                r = getrandbits(k)                while r >= n:                    r = getrandbits(k)                return r        if n >= _maxwidth:            _warn("Underlying random() generator does not supply \n"                "enough bits to choose from a population range this large")        return int(self.random() * n) ## -------------------- sequence methods  -------------------     def choice(self, seq):        """Choose a random element from a non-empty sequence."""        return seq[int(self.random() * len(seq))]  # raises IndexError if seq is empty     def shuffle(self, x, random=None, int=int):        """x, random=random.random -> shuffle list x in place; return None.         Optional arg random is a 0-argument function returning a random        float in [0.0, 1.0); by default, the standard random.random.        """         if random is None:            random = self.random        for i in reversed(xrange(1, len(x))):            # pick an element in x[:i+1] with which to exchange x[i]            j = int(random() * (i+1))            x[i], x[j] = x[j], x[i]     def sample(self, population, k):        """Chooses k unique random elements from a population sequence.         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 in 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)        """         # XXX Although the documentation says `population` is "a sequence",        # XXX attempts are made to cater to any iterable with a __len__        # XXX method.  This has had mixed success.  Examples from both        # XXX sides:  sets work fine, and should become officially supported;        # XXX dicts are much harder, and have failed in various subtle        # XXX ways across attempts.  Support for mapping types should probably        # XXX be dropped (and users should pass mapping.keys() or .values()        # XXX explicitly).         # Sampling without replacement entails tracking either potential        # selections (the pool) in a list or previous selections in a set.         # When the number of selections is small compared to the        # population, then tracking selections is efficient, requiring        # only a small set and an occasional reselection.  For        # a larger number of selections, the pool tracking method is        # preferred since the list takes less space than the        # set and it doesn't suffer from frequent reselections.         n = len(population)        if not 0 <= k <= n:            raise ValueError, "sample larger than population"        random = self.random        _int = int        result = [None] * k        setsize = 21        # size of a small set minus size of an empty list        if k > 5:            setsize += 4 ** _ceil(_log(k * 3, 4)) # table size for big sets        if n <= setsize or hasattr(population, "keys"):            # An n-length list is smaller than a k-length set, or this is a            # mapping type so the other algorithm wouldn't work.            pool = list(population)            for i in xrange(k):         # invariant:  non-selected at [0,n-i)                j = _int(random() * (n-i))                result[i] = pool[j]                pool[j] = pool[n-i-1]   # move non-selected item into vacancy        else:            try:                selected = set()                selected_add = selected.add                for i in xrange(k):                    j = _int(random() * n)                    while j in selected:                        j = _int(random() * n)                    selected_add(j)                    result[i] = population[j]            except (TypeError, KeyError):   # handle (at least) sets                if isinstance(population, list):                    raise                return self.sample(tuple(population), k)        return result ## -------------------- real-valued distributions  ------------------- ## -------------------- uniform distribution -------------------     def uniform(self, a, b):        """Get a random number in the range [a, b)."""        return a + (b-a) * self.random() ## -------------------- normal distribution --------------------     def normalvariate(self, mu, sigma):        """Normal distribution.         mu is the mean, and sigma is the standard deviation.         """        # mu = mean, sigma = standard deviation         # Uses Kinderman and Monahan method. Reference: Kinderman,        # A.J. and Monahan, J.F., "Computer generation of random        # variables using the ratio of uniform deviates", ACM Trans        # Math Software, 3, (1977), pp257-260.         random = self.random        while 1:            u1 = random()            u2 = 1.0 - random()            z = NV_MAGICCONST*(u1-0.5)/u2            zz = z*z/4.0            if zz <= -_log(u2):                break        return mu + z*sigma ## -------------------- lognormal distribution --------------------     def lognormvariate(self, 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.         """        return _exp(self.normalvariate(mu, sigma)) ## -------------------- exponential distribution --------------------     def expovariate(self, 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.         """        # lambd: rate lambd = 1/mean        # ('lambda' is a Python reserved word)         random = self.random        u = random()        while u <= 1e-7:            u = random()        return -_log(u)/lambd ## -------------------- von Mises distribution --------------------     def vonmisesvariate(self, mu, kappa):        """Circular data distribution.         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.         """        # mu:    mean angle (in radians between 0 and 2*pi)        # kappa: concentration parameter kappa (>= 0)        # if kappa = 0 generate uniform random angle         # Based upon an algorithm published in: Fisher, N.I.,        # "Statistical Analysis of Circular Data", Cambridge        # University Press, 1993.         # Thanks to Magnus Kessler for a correction to the        # implementation of step 4.         random = self.random        if kappa <= 1e-6:            return TWOPI * random()         a = 1.0 + _sqrt(1.0 + 4.0 * kappa * kappa)        b = (a - _sqrt(2.0 * a))/(2.0 * kappa)        r = (1.0 + b * b)/(2.0 * b)         while 1:            u1 = random()             z = _cos(_pi * u1)            f = (1.0 + r * z)/(r + z)            c = kappa * (r - f)             u2 = random()             if u2 < c * (2.0 - c) or u2 <= c * _exp(1.0 - c):                break         u3 = random()        if u3 > 0.5:            theta = (mu % TWOPI) + _acos(f)        else:            theta = (mu % TWOPI) - _acos(f)         return theta ## -------------------- gamma distribution --------------------     def gammavariate(self, alpha, beta):        """Gamma distribution.  Not the gamma function!         Conditions on the parameters are alpha > 0 and beta > 0.         """         # alpha > 0, beta > 0, mean is alpha*beta, variance is alpha*beta**2         # Warning: a few older sources define the gamma distribution in terms        # of alpha > -1.0        if alpha <= 0.0 or beta <= 0.0:            raise ValueError, 'gammavariate: alpha and beta must be > 0.0'         random = self.random        if alpha > 1.0:             # Uses R.C.H. Cheng, "The generation of Gamma            # variables with non-integral shape parameters",            # Applied Statistics, (1977), 26, No. 1, p71-74             ainv = _sqrt(2.0 * alpha - 1.0)            bbb = alpha - LOG4            ccc = alpha + ainv             while 1:                u1 = random()                if not 1e-7 < u1 < .9999999:                    continue                u2 = 1.0 - random()                v = _log(u1/(1.0-u1))/ainv                x = alpha*_exp(v)                z = u1*u1*u2                r = bbb+ccc*v-x                if r + SG_MAGICCONST - 4.5*z >= 0.0 or r >= _log(z):                    return x * beta         elif alpha == 1.0:            # expovariate(1)            u = random()            while u <= 1e-7:                u = random()            return -_log(u) * beta         else:   # alpha is between 0 and 1 (exclusive)             # Uses ALGORITHM GS of Statistical Computing - Kennedy & Gentle             while 1:                u = random()                b = (_e + alpha)/_e

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