code: #
  # Author:  Travis Oliphant  2002-2011 with contributions from
  #          SciPy Developers 2004-2011
  #
  from __future__ import division, print_function, absolute_import
  
  from scipy._lib.six import string_types, exec_
  from scipy._lib._util import getargspec_no_self as _getargspec
  
  import sys
  import keyword
  import re
  import types
  import warnings
  
  from scipy.misc import doccer
  from ._distr_params import distcont, distdiscrete
  from scipy._lib._util import check_random_state, _lazywhere
  from scipy._lib._util import _valarray as valarray
  
  from scipy.special import (comb, chndtr, gammaln, entr, kl_div, xlogy, ive)
  
  # for root finding for discrete distribution ppf, and max likelihood estimation
  from scipy import optimize
  
  # for functions of continuous distributions (e.g. moments, entropy, cdf)
  from scipy import integrate
  
  # to approximate the pdf of a continuous distribution given its cdf
  from scipy.misc import derivative
  
  from numpy import (arange, putmask, ravel, take, ones, shape, ndarray,
                     product, reshape, zeros, floor, logical_and, log, sqrt, exp)
  
  from numpy import (place, argsort, argmax, vectorize,
                     asarray, nan, inf, isinf, NINF, empty)
  
  import numpy as np
  
  from ._constants import _EPS, _XMAX
  
  try:
      from new import instancemethod
  except ImportError:
      # Python 3
      def instancemethod(func, obj, cls):
          return types.MethodType(func, obj)
  
  
  # These are the docstring parts used for substitution in specific
  # distribution docstrings
  
  docheaders = {'methods': """\nMethods\n-------\n""",
                'notes': """\nNotes\n-----\n""",
                'examples': """\nExamples\n--------\n"""}
  
  _doc_rvs = """\
  ``rvs(%(shapes)s, loc=0, scale=1, size=1, random_state=None)``
      Random variates.
  """
  _doc_pdf = """\
  ``pdf(x, %(shapes)s, loc=0, scale=1)``
      Probability density function.
  """
  _doc_logpdf = """\
  ``logpdf(x, %(shapes)s, loc=0, scale=1)``
      Log of the probability density function.
  """
  _doc_pmf = """\
  ``pmf(x, %(shapes)s, loc=0, scale=1)``
      Probability mass function.
  """
  _doc_logpmf = """\
  ``logpmf(x, %(shapes)s, loc=0, scale=1)``
      Log of the probability mass function.
  """
  _doc_cdf = """\
  ``cdf(x, %(shapes)s, loc=0, scale=1)``
      Cumulative density function.
  """
  _doc_logcdf = """\
  ``logcdf(x, %(shapes)s, loc=0, scale=1)``
      Log of the cumulative density function.
  """
  _doc_sf = """\
  ``sf(x, %(shapes)s, loc=0, scale=1)``
      Survival function  (also defined as ``1 - cdf``, but `sf` is sometimes more accurate).
  """
  _doc_logsf = """\
  ``logsf(x, %(shapes)s, loc=0, scale=1)``
      Log of the survival function.
  """
  _doc_ppf = """\
  ``ppf(q, %(shapes)s, loc=0, scale=1)``
      Percent point function (inverse of ``cdf`` --- percentiles).
  """
  _doc_isf = """\
  ``isf(q, %(shapes)s, loc=0, scale=1)``
      Inverse survival function (inverse of ``sf``).
  """
  _doc_moment = """\
  ``moment(n, %(shapes)s, loc=0, scale=1)``
      Non-central moment of order n
  """
  _doc_stats = """\
  ``stats(%(shapes)s, loc=0, scale=1, moments='mv')``
      Mean('m'), variance('v'), skew('s'), and/or kurtosis('k').
  """
  _doc_entropy = """\
  ``entropy(%(shapes)s, loc=0, scale=1)``
      (Differential) entropy of the RV.
  """
  _doc_fit = """\
  ``fit(data, %(shapes)s, loc=0, scale=1)``
      Parameter estimates for generic data.
  """
  _doc_expect = """\
  ``expect(func, args=(%(shapes_)s), loc=0, scale=1, lb=None, ub=None, conditional=False, **kwds)``
      Expected value of a function (of one argument) with respect to the distribution.
  """
  _doc_expect_discrete = """\
  ``expect(func, args=(%(shapes_)s), loc=0, lb=None, ub=None, conditional=False)``
      Expected value of a function (of one argument) with respect to the distribution.
  """
  _doc_median = """\
  ``median(%(shapes)s, loc=0, scale=1)``
      Median of the distribution.
  """
  _doc_mean = """\
  ``mean(%(shapes)s, loc=0, scale=1)``
      Mean of the distribution.
  """
  _doc_var = """\
  ``var(%(shapes)s, loc=0, scale=1)``
      Variance of the distribution.
  """
  _doc_std = """\
  ``std(%(shapes)s, loc=0, scale=1)``
      Standard deviation of the distribution.
  """
  _doc_interval = """\
  ``interval(alpha, %(shapes)s, loc=0, scale=1)``
      Endpoints of the range that contains alpha percent of the distribution
  """
  _doc_allmethods = ''.join([docheaders['methods'], _doc_rvs, _doc_pdf,
                             _doc_logpdf, _doc_cdf, _doc_logcdf, _doc_sf,
                             _doc_logsf, _doc_ppf, _doc_isf, _doc_moment,
                             _doc_stats, _doc_entropy, _doc_fit,
                             _doc_expect, _doc_median,
                             _doc_mean, _doc_var, _doc_std, _doc_interval])
  
  _doc_default_longsummary = """\
  As an instance of the `rv_continuous` class, `%(name)s` object inherits from it
  a collection of generic methods (see below for the full list),
  and completes them with details specific for this particular distribution.
  """
  
  _doc_default_frozen_note = """
  Alternatively, the object may be called (as a function) to fix the shape,
  location, and scale parameters returning a "frozen" continuous RV object:
  
  rv = %(name)s(%(shapes)s, loc=0, scale=1)
      - Frozen RV object with the same methods but holding the given shape,
        location, and scale fixed.
  """
  _doc_default_example = """\
  Examples
  --------
  >>> from scipy.stats import %(name)s
  >>> import matplotlib.pyplot as plt
  >>> fig, ax = plt.subplots(1, 1)
  
  Calculate a few first moments:
  
  %(set_vals_stmt)s
  >>> mean, var, skew, kurt = %(name)s.stats(%(shapes)s, moments='mvsk')
  
  Display the probability density function (``pdf``):
  
  >>> x = np.linspace(%(name)s.ppf(0.01, %(shapes)s),
  ...                 %(name)s.ppf(0.99, %(shapes)s), 100)
  >>> ax.plot(x, %(name)s.pdf(x, %(shapes)s),
  ...        'r-', lw=5, alpha=0.6, label='%(name)s pdf')
  
  Alternatively, the distribution object can be called (as a function)
  to fix the shape, location and scale parameters. This returns a "frozen"
  RV object holding the given parameters fixed.
  
  Freeze the distribution and display the frozen ``pdf``:
  
  >>> rv = %(name)s(%(shapes)s)
  >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')
  
  Check accuracy of ``cdf`` and ``ppf``:
  
  >>> vals = %(name)s.ppf([0.001, 0.5, 0.999], %(shapes)s)
  >>> np.allclose([0.001, 0.5, 0.999], %(name)s.cdf(vals, %(shapes)s))
  True
  
  Generate random numbers:
  
  >>> r = %(name)s.rvs(%(shapes)s, size=1000)
  
  And compare the histogram:
  
  >>> ax.hist(r, normed=True, histtype='stepfilled', alpha=0.2)
  >>> ax.legend(loc='best', frameon=False)
  >>> plt.show()
  
  """
  
  _doc_default_locscale = """\
  The probability density above is defined in the "standardized" form. To shift
  and/or scale the distribution use the ``loc`` and ``scale`` parameters.
  Specifically, ``%(name)s.pdf(x, %(shapes)s, loc, scale)`` is identically
  equivalent to ``%(name)s.pdf(y, %(shapes)s) / scale`` with
  ``y = (x - loc) / scale``.
  """
  
  _doc_default = ''.join([_doc_default_longsummary,
                          _doc_allmethods,
                          '\n',
                          _doc_default_example])
  
  _doc_default_before_notes = ''.join([_doc_default_longsummary,
                                       _doc_allmethods])
  
  docdict = {
      'rvs': _doc_rvs,
      'pdf': _doc_pdf,
      'logpdf': _doc_logpdf,
      'cdf': _doc_cdf,
      'logcdf': _doc_logcdf,
      'sf': _doc_sf,
      'logsf': _doc_logsf,
      'ppf': _doc_ppf,
      'isf': _doc_isf,
      'stats': _doc_stats,
      'entropy': _doc_entropy,
      'fit': _doc_fit,
      'moment': _doc_moment,
      'expect': _doc_expect,
      'interval': _doc_interval,
      'mean': _doc_mean,
      'std': _doc_std,
      'var': _doc_var,
      'median': _doc_median,
      'allmethods': _doc_allmethods,
      'longsummary': _doc_default_longsummary,
      'frozennote': _doc_default_frozen_note,
      'example': _doc_default_example,
      'default': _doc_default,
      'before_notes': _doc_default_before_notes,
      'after_notes': _doc_default_locscale
  }
  
  # Reuse common content between continuous and discrete docs, change some
  # minor bits.
  docdict_discrete = docdict.copy()
  
  docdict_discrete['pmf'] = _doc_pmf
  docdict_discrete['logpmf'] = _doc_logpmf
  docdict_discrete['expect'] = _doc_expect_discrete
  _doc_disc_methods = ['rvs', 'pmf', 'logpmf', 'cdf', 'logcdf', 'sf', 'logsf',
                       'ppf', 'isf', 'stats', 'entropy', 'expect', 'median',
                       'mean', 'var', 'std', 'interval']
  for obj in _doc_disc_methods:
      docdict_discrete[obj] = docdict_discrete[obj].replace(', scale=1', '')
  docdict_discrete.pop('pdf')
  docdict_discrete.pop('logpdf')
  
  _doc_allmethods = ''.join([docdict_discrete[obj] for obj in _doc_disc_methods])
  docdict_discrete['allmethods'] = docheaders['methods'] + _doc_allmethods
  
  docdict_discrete['longsummary'] = _doc_default_longsummary.replace(
      'rv_continuous', 'rv_discrete')
  
  _doc_default_frozen_note = """
  Alternatively, the object may be called (as a function) to fix the shape and
  location parameters returning a "frozen" discrete RV object:
  
  rv = %(name)s(%(shapes)s, loc=0)
      - Frozen RV object with the same methods but holding the given shape and
        location fixed.
  """
  docdict_discrete['frozennote'] = _doc_default_frozen_note
  
  _doc_default_discrete_example = """\
  Examples
  --------
  >>> from scipy.stats import %(name)s
  >>> import matplotlib.pyplot as plt
  >>> fig, ax = plt.subplots(1, 1)
  
  Calculate a few first moments:
  
  %(set_vals_stmt)s
  >>> mean, var, skew, kurt = %(name)s.stats(%(shapes)s, moments='mvsk')
  
  Display the probability mass function (``pmf``):
  
  >>> x = np.arange(%(name)s.ppf(0.01, %(shapes)s),
  ...               %(name)s.ppf(0.99, %(shapes)s))
  >>> ax.plot(x, %(name)s.pmf(x, %(shapes)s), 'bo', ms=8, label='%(name)s pmf')
  >>> ax.vlines(x, 0, %(name)s.pmf(x, %(shapes)s), colors='b', lw=5, alpha=0.5)
  
  Alternatively, the distribution object can be called (as a function)
  to fix the shape and location. This returns a "frozen" RV object holding
  the given parameters fixed.
  
  Freeze the distribution and display the frozen ``pmf``:
  
  >>> rv = %(name)s(%(shapes)s)
  >>> ax.vlines(x, 0, rv.pmf(x), colors='k', linestyles='-', lw=1,
  ...         label='frozen pmf')
  >>> ax.legend(loc='best', frameon=False)
  >>> plt.show()
  
  Check accuracy of ``cdf`` and ``ppf``:
  
  >>> prob = %(name)s.cdf(x, %(shapes)s)
  >>> np.allclose(x, %(name)s.ppf(prob, %(shapes)s))
  True
  
  Generate random numbers:
  
  >>> r = %(name)s.rvs(%(shapes)s, size=1000)
  """
  
  
  _doc_default_discrete_locscale = """\
  The probability mass function above is defined in the "standardized" form.
  To shift distribution use the ``loc`` parameter.
  Specifically, ``%(name)s.pmf(k, %(shapes)s, loc)`` is identically
  equivalent to ``%(name)s.pmf(k - loc, %(shapes)s)``.
  """
  
  docdict_discrete['example'] = _doc_default_discrete_example
  docdict_discrete['after_notes'] = _doc_default_discrete_locscale
  
  _doc_default_before_notes = ''.join([docdict_discrete['longsummary'],
                                       docdict_discrete['allmethods']])
  docdict_discrete['before_notes'] = _doc_default_before_notes
  
  _doc_default_disc = ''.join([docdict_discrete['longsummary'],
                               docdict_discrete['allmethods'],
                               docdict_discrete['frozennote'],
                               docdict_discrete['example']])
  docdict_discrete['default'] = _doc_default_disc
  
  # clean up all the separate docstring elements, we do not need them anymore
  for obj in [s for s in dir() if s.startswith('_doc_')]:
      exec('del ' + obj)
  del obj
  try:
      del s
  except NameError:
      # in Python 3, loop variables are not visible after the loop
      pass
  
  
  def _moment(data, n, mu=None):
      if mu is None:
          mu = data.mean()
      return ((data - mu)**n).mean()
  
  
  def _moment_from_stats(n, mu, mu2, g1, g2, moment_func, args):
      if (n == 0):
          return 1.0
      elif (n == 1):
          if mu is None:
              val = moment_func(1, *args)
          else:
              val = mu
      elif (n == 2):
          if mu2 is None or mu is None:
              val = moment_func(2, *args)
          else:
              val = mu2 + mu*mu
      elif (n == 3):
          if g1 is None or mu2 is None or mu is None:
              val = moment_func(3, *args)
          else:
              mu3 = g1 * np.power(mu2, 1.5)  # 3rd central moment
              val = mu3+3*mu*mu2+mu*mu*mu  # 3rd non-central moment
      elif (n == 4):
          if g1 is None or g2 is None or mu2 is None or mu is None:
              val = moment_func(4, *args)
          else:
              mu4 = (g2+3.0)*(mu2**2.0)  # 4th central moment
              mu3 = g1*np.power(mu2, 1.5)  # 3rd central moment
              val = mu4+4*mu*mu3+6*mu*mu*mu2+mu*mu*mu*mu
      else:
          val = moment_func(n, *args)
  
      return val
  
  
  def _skew(data):
      """
      skew is third central moment / variance**(1.5)
      """
      data = np.ravel(data)
      mu = data.mean()
      m2 = ((data - mu)**2).mean()
      m3 = ((data - mu)**3).mean()
      return m3 / np.power(m2, 1.5)
  
  
  def _kurtosis(data):
      """
      kurtosis is fourth central moment / variance**2 - 3
      """
      data = np.ravel(data)
      mu = data.mean()
      m2 = ((data - mu)**2).mean()
      m4 = ((data - mu)**4).mean()
      return m4 / m2**2 - 3
  
  
  # Frozen RV class
  class rv_frozen(object):
  
      def __init__(self, dist, *args, **kwds):
          self.args = args
          self.kwds = kwds
  
          # create a new instance
          self.dist = dist.__class__(**dist._updated_ctor_param())
  
          # a, b may be set in _argcheck, depending on *args, **kwds. Ouch.
          shapes, _, _ = self.dist._parse_args(*args, **kwds)
          self.dist._argcheck(*shapes)
          self.a, self.b = self.dist.a, self.dist.b
  
      @property
      def random_state(self):
          return self.dist._random_state
  
      @random_state.setter
      def random_state(self, seed):
          self.dist._random_state = check_random_state(seed)
  
      def pdf(self, x):   # raises AttributeError in frozen discrete distribution
          return self.dist.pdf(x, *self.args, **self.kwds)
  
      def logpdf(self, x):
          return self.dist.logpdf(x, *self.args, **self.kwds)
  
      def cdf(self, x):
          return self.dist.cdf(x, *self.args, **self.kwds)
  
      def logcdf(self, x):
          return self.dist.logcdf(x, *self.args, **self.kwds)
  
      def ppf(self, q):
          return self.dist.ppf(q, *self.args, **self.kwds)
  
      def isf(self, q):
          return self.dist.isf(q, *self.args, **self.kwds)
  
      def rvs(self, size=None, random_state=None):
          kwds = self.kwds.copy()
          kwds.update({'size': size, 'random_state': random_state})
          return self.dist.rvs(*self.args, **kwds)
  
      def sf(self, x):
          return self.dist.sf(x, *self.args, **self.kwds)
  
      def logsf(self, x):
          return self.dist.logsf(x, *self.args, **self.kwds)
  
      def stats(self, moments='mv'):
          kwds = self.kwds.copy()
          kwds.update({'moments': moments})
          return self.dist.stats(*self.args, **kwds)
  
      def median(self):
          return self.dist.median(*self.args, **self.kwds)
  
      def mean(self):
          return self.dist.mean(*self.args, **self.kwds)
  
      def var(self):
          return self.dist.var(*self.args, **self.kwds)
  
      def std(self):
          return self.dist.std(*self.args, **self.kwds)
  
      def moment(self, n):
          return self.dist.moment(n, *self.args, **self.kwds)
  
      def entropy(self):
          return self.dist.entropy(*self.args, **self.kwds)
  
      def pmf(self, k):
          return self.dist.pmf(k, *self.args, **self.kwds)
  
      def logpmf(self, k):
          return self.dist.logpmf(k, *self.args, **self.kwds)
  
      def interval(self, alpha):
          return self.dist.interval(alpha, *self.args, **self.kwds)
  
      def expect(self, func=None, lb=None, ub=None, conditional=False, **kwds):
          # expect method only accepts shape parameters as positional args
          # hence convert self.args, self.kwds, also loc/scale
          # See the .expect method docstrings for the meaning of
          # other parameters.
          a, loc, scale = self.dist._parse_args(*self.args, **self.kwds)
          if isinstance(self.dist, rv_discrete):
              return self.dist.expect(func, a, loc, lb, ub, conditional, **kwds)
          else:
              return self.dist.expect(func, a, loc, scale, lb, ub,
                                      conditional, **kwds)
  
  
  # This should be rewritten
  def argsreduce(cond, *args):
      """Return the sequence of ravel(args[i]) where ravel(condition) is
      True in 1D.
  
      Examples
      --------
      >>> import numpy as np
      >>> rand = np.random.random_sample
      >>> A = rand((4, 5))
      >>> B = 2
      >>> C = rand((1, 5))
      >>> cond = np.ones(A.shape)
      >>> [A1, B1, C1] = argsreduce(cond, A, B, C)
      >>> B1.shape
      (20,)
      >>> cond[2,:] = 0
      >>> [A2, B2, C2] = argsreduce(cond, A, B, C)
      >>> B2.shape
      (15,)
  
      """
      newargs = np.atleast_1d(*args)
      if not isinstance(newargs, list):
          newargs = [newargs, ]
      expand_arr = (cond == cond)
      return [np.extract(cond, arr1 * expand_arr) for arr1 in newargs]
  
  
  parse_arg_template = """
  def _parse_args(self, %(shape_arg_str)s %(locscale_in)s):
      return (%(shape_arg_str)s), %(locscale_out)s
  
  def _parse_args_rvs(self, %(shape_arg_str)s %(locscale_in)s, size=None):
      return (%(shape_arg_str)s), %(locscale_out)s, size
  
  def _parse_args_stats(self, %(shape_arg_str)s %(locscale_in)s, moments='mv'):
      return (%(shape_arg_str)s), %(locscale_out)s, moments
  """
  
  
  # Both the continuous and discrete distributions depend on ncx2.
  # I think the function name ncx2 is an abbreviation for noncentral chi squared.
  
  def _ncx2_log_pdf(x, df, nc):
      # We use (xs**2 + ns**2)/2 = (xs - ns)**2/2  + xs*ns, and include the factor
      # of exp(-xs*ns) into the ive function to improve numerical stability
      # at large values of xs. See also `rice.pdf`.
      df2 = df/2.0 - 1.0
      xs, ns = np.sqrt(x), np.sqrt(nc)
      res = xlogy(df2/2.0, x/nc) - 0.5*(xs - ns)**2
      res += np.log(ive(df2, xs*ns) / 2.0)
      return res
  
  
  def _ncx2_pdf(x, df, nc):
      return np.exp(_ncx2_log_pdf(x, df, nc))
  
  
  def _ncx2_cdf(x, df, nc):
      return chndtr(x, df, nc)
  
  
  class rv_generic(object):
      """Class which encapsulates common functionality between rv_discrete
      and rv_continuous.
  
      """
      def __init__(self, seed=None):
          super(rv_generic, self).__init__()
  
          # figure out if _stats signature has 'moments' keyword
          sign = _getargspec(self._stats)
          self._stats_has_moments = ((sign[2] is not None) or
                                     ('moments' in sign[0]))
          self._random_state = check_random_state(seed)
  
      @property
      def random_state(self):
          """ Get or set the RandomState object for generating random variates.
  
          This can be either None or an existing RandomState object.
  
          If None (or np.random), use the RandomState singleton used by np.random.
          If already a RandomState instance, use it.
          If an int, use a new RandomState instance seeded with seed.
  
          """
          return self._random_state
  
      @random_state.setter
      def random_state(self, seed):
          self._random_state = check_random_state(seed)
  
      def __getstate__(self):
          return self._updated_ctor_param(), self._random_state
  
      def __setstate__(self, state):
          ctor_param, r = state
          self.__init__(**ctor_param)
          self._random_state = r
          return self
  
      def _construct_argparser(
              self, meths_to_inspect, locscale_in, locscale_out):
          """Construct the parser for the shape arguments.
  
          Generates the argument-parsing functions dynamically and attaches
          them to the instance.
          Is supposed to be called in __init__ of a class for each distribution.
  
          If self.shapes is a non-empty string, interprets it as a
          comma-separated list of shape parameters.
  
          Otherwise inspects the call signatures of `meths_to_inspect`
          and constructs the argument-parsing functions from these.
          In this case also sets `shapes` and `numargs`.
          """
  
          if self.shapes:
              # sanitize the user-supplied shapes
              if not isinstance(self.shapes, string_types):
                  raise TypeError('shapes must be a string.')
  
              shapes = self.shapes.replace(',', ' ').split()
  
              for field in shapes:
                  if keyword.iskeyword(field):
                      raise SyntaxError('keywords cannot be used as shapes.')
                  if not re.match('^[_a-zA-Z][_a-zA-Z0-9]*$', field):
                      raise SyntaxError(
                          'shapes must be valid python identifiers')
          else:
              # find out the call signatures (_pdf, _cdf etc), deduce shape
              # arguments. Generic methods only have 'self, x', any further args
              # are shapes.
              shapes_list = []
              for meth in meths_to_inspect:
                  shapes_args = _getargspec(meth)   # NB: does not contain self
                  args = shapes_args.args[1:]       # peel off 'x', too
  
                  if args:
                      shapes_list.append(args)
  
                      # *args or **kwargs are not allowed w/automatic shapes
                      if shapes_args.varargs is not None:
                          raise TypeError(
                              '*args are not allowed w/out explicit shapes')
                      if shapes_args.keywords is not None:
                          raise TypeError(
                              '**kwds are not allowed w/out explicit shapes')
                      if shapes_args.defaults is not None:
                          raise TypeError('defaults are not allowed for shapes')
  
              if shapes_list:
                  shapes = shapes_list[0]
  
                  # make sure the signatures are consistent
                  for item in shapes_list:
                      if item != shapes:
                          raise TypeError('Shape arguments are inconsistent.')
              else:
                  shapes = []
  
          # have the arguments, construct the method from template
          shapes_str = ', '.join(shapes) + ', ' if shapes else ''  # NB: not None
          dct = dict(shape_arg_str=shapes_str,
                     locscale_in=locscale_in,
                     locscale_out=locscale_out,
                     )
          ns = {}
          exec_(parse_arg_template % dct, ns)
          # NB: attach to the instance, not class
          for name in ['_parse_args', '_parse_args_stats', '_parse_args_rvs']:
              setattr(self, name,
                      instancemethod(ns[name], self, self.__class__)
                      )
  
          self.shapes = ', '.join(shapes) if shapes else None
          if not hasattr(self, 'numargs'):
              # allows more general subclassing with *args
              self.numargs = len(shapes)
  
      def _construct_doc(self, docdict, shapes_vals=None):
          """Construct the instance docstring with string substitutions."""
          tempdict = docdict.copy()
          tempdict['name'] = self.name or 'distname'
          tempdict['shapes'] = self.shapes or ''
  
          if shapes_vals is None:
              shapes_vals = ()
          vals = ', '.join('%.3g' % val for val in shapes_vals)
          tempdict['vals'] = vals
  
          tempdict['shapes_'] = self.shapes or ''
          if self.shapes and self.numargs == 1:
              tempdict['shapes_'] += ','
  
          if self.shapes:
              tempdict['set_vals_stmt'] = '>>> %s = %s' % (self.shapes, vals)
          else:
              tempdict['set_vals_stmt'] = ''
  
          if self.shapes is None:
              # remove shapes from call parameters if there are none
              for item in ['default', 'before_notes']:
                  tempdict[item] = tempdict[item].replace(
                      "\n%(shapes)s : array_like\n    shape parameters", "")
          for i in range(2):
              if self.shapes is None:
                  # necessary because we use %(shapes)s in two forms (w w/o ", ")
                  self.__doc__ = self.__doc__.replace("%(shapes)s, ", "")
              self.__doc__ = doccer.docformat(self.__doc__, tempdict)
  
          # correct for empty shapes
          self.__doc__ = self.__doc__.replace('(, ', '(').replace(', )', ')')
  
      def _construct_default_doc(self, longname=None, extradoc=None,
                  docdict=None, discrete='continuous'):
          """Construct instance docstring from the default template."""
          if longname is None:
              longname = 'A'
          if extradoc is None:
              extradoc = ''
          if extradoc.startswith('\n\n'):
              extradoc = extradoc[2:]
          self.__doc__ = ''.join(['%s %s random variable.' % (longname, discrete),
                                  '\n\n%(before_notes)s\n', docheaders['notes'],
                                  extradoc, '\n%(example)s'])
          self._construct_doc(docdict)
  
      def freeze(self, *args, **kwds):
          """Freeze the distribution for the given arguments.
  
          Parameters
          ----------
          arg1, arg2, arg3,... : array_like
              The shape parameter(s) for the distribution.  Should include all
              the non-optional arguments, may include ``loc`` and ``scale``.
  
          Returns
          -------
          rv_frozen : rv_frozen instance
              The frozen distribution.
  
          """
          return rv_frozen(self, *args, **kwds)
  
      def __call__(self, *args, **kwds):
          return self.freeze(*args, **kwds)
      __call__.__doc__ = freeze.__doc__
  
      # The actual calculation functions (no basic checking need be done)
      # If these are defined, the others won't be looked at.
      # Otherwise, the other set can be defined.
      def _stats(self, *args, **kwds):
          return None, None, None, None
  
      #  Central moments
      def _munp(self, n, *args):
          # Silence floating point warnings from integration.
          olderr = np.seterr(all='ignore')
          vals = self.generic_moment(n, *args)
          np.seterr(**olderr)
          return vals
  
      ## These are the methods you must define (standard form functions)
      ## NB: generic _pdf, _logpdf, _cdf are different for
      ## rv_continuous and rv_discrete hence are defined in there
      def _argcheck(self, *args):
          """Default check for correct values on args and keywords.
  
          Returns condition array of 1's where arguments are correct and
           0's where they are not.
  
          """
          cond = 1
          for arg in args:
              cond = logical_and(cond, (asarray(arg) > 0))
          return cond
  
      ##(return 1-d using self._size to get number)
      def _rvs(self, *args):
          ## Use basic inverse cdf algorithm for RV generation as default.
          U = self._random_state.random_sample(self._size)
          Y = self._ppf(U, *args)
          return Y
  
      def _logcdf(self, x, *args):
          return log(self._cdf(x, *args))
  
      def _sf(self, x, *args):
          return 1.0-self._cdf(x, *args)
  
      def _logsf(self, x, *args):
          return log(self._sf(x, *args))
  
      def _ppf(self, q, *args):
          return self._ppfvec(q, *args)
  
      def _isf(self, q, *args):
          return self._ppf(1.0-q, *args)  # use correct _ppf for subclasses
  
      # These are actually called, and should not be overwritten if you
      # want to keep error checking.
      def rvs(self, *args, **kwds):
          """
          Random variates of given type.
  
          Parameters
          ----------
          arg1, arg2, arg3,... : array_like
              The shape parameter(s) for the distribution (see docstring of the
              instance object for more information).
          loc : array_like, optional
              Location parameter (default=0).
          scale : array_like, optional
              Scale parameter (default=1).
          size : int or tuple of ints, optional
              Defining number of random variates (default is 1).
          random_state : None or int or ``np.random.RandomState`` instance, optional
              If int or RandomState, use it for drawing the random variates.
              If None, rely on ``self.random_state``.
              Default is None.
  
          Returns
          -------
          rvs : ndarray or scalar
              Random variates of given `size`.
  
          """
          discrete = kwds.pop('discrete', None)
          rndm = kwds.pop('random_state', None)
          args, loc, scale, size = self._parse_args_rvs(*args, **kwds)
          cond = logical_and(self._argcheck(*args), (scale >= 0))
          if not np.all(cond):
              raise ValueError("Domain error in arguments.")
  
          # self._size is total size of all output values
          self._size = product(size, axis=0)
          if self._size is not None and self._size > 1:
              size = np.array(size, ndmin=1)
  
          if np.all(scale == 0):
              return loc*ones(size, 'd')
  
          # extra gymnastics needed for a custom random_state
          if rndm is not None:
              random_state_saved = self._random_state
              self._random_state = check_random_state(rndm)
  
          vals = self._rvs(*args)
          if self._size is not None:
              vals = reshape(vals, size)
  
          vals = vals * scale + loc
  
          # do not forget to restore the _random_state
          if rndm is not None:
              self._random_state = random_state_saved
  
          # Cast to int if discrete
          if discrete:
              if np.isscalar(vals):
                  vals = int(vals)
              else:
                  vals = vals.astype(int)
  
          return vals
  
      def stats(self, *args, **kwds):
          """
          Some statistics of the given RV.
  
          Parameters
          ----------
          arg1, arg2, arg3,... : array_like
              The shape parameter(s) for the distribution (see docstring of the
              instance object for more information)
          loc : array_like, optional
              location parameter (default=0)
          scale : array_like, optional (continuous RVs only)
              scale parameter (default=1)
          moments : str, optional
              composed of letters ['mvsk'] defining which moments to compute:
              'm' = mean,
              'v' = variance,
              's' = (Fisher's) skew,
              'k' = (Fisher's) kurtosis.
              (default is 'mv')
  
          Returns
          -------
          stats : sequence
              of requested moments.
  
          """
          args, loc, scale, moments = self._parse_args_stats(*args, **kwds)
          # scale = 1 by construction for discrete RVs
          loc, scale = map(asarray, (loc, scale))
          args = tuple(map(asarray, args))
          cond = self._argcheck(*args) & (scale > 0) & (loc == loc)
          output = []
          default = valarray(shape(cond), self.badvalue)
  
          # Use only entries that are valid in calculation
          if np.any(cond):
              goodargs = argsreduce(cond, *(args+(scale, loc)))
              scale, loc, goodargs = goodargs[-2], goodargs[-1], goodargs[:-2]
  
              if self._stats_has_moments:
                  mu, mu2, g1, g2 = self._stats(*goodargs,
                                                **{'moments': moments})
              else:
                  mu, mu2, g1, g2 = self._stats(*goodargs)
              if g1 is None:
                  mu3 = None
              else:
                  if mu2 is None:
                      mu2 = self._munp(2, *goodargs)
                  if g2 is None:
                      # (mu2**1.5) breaks down for nan and inf
                      mu3 = g1 * np.power(mu2, 1.5)
  
              if 'm' in moments:
                  if mu is None:
                      mu = self._munp(1, *goodargs)
                  out0 = default.copy()
                  place(out0, cond, mu * scale + loc)
                  output.append(out0)
  
              if 'v' in moments:
                  if mu2 is None:
                      mu2p = self._munp(2, *goodargs)
                      if mu is None:
                          mu = self._munp(1, *goodargs)
                      mu2 = mu2p - mu * mu
                      if np.isinf(mu):
                          #if mean is inf then var is also inf
                          mu2 = np.inf
                  out0 = default.copy()
                  place(out0, cond, mu2 * scale * scale)
                  output.append(out0)
  
              if 's' in moments:
                  if g1 is None:
                      mu3p = self._munp(3, *goodargs)
                      if mu is None:
                          mu = self._munp(1, *goodargs)
                      if mu2 is None:
                          mu2p = self._munp(2, *goodargs)
                          mu2 = mu2p - mu * mu
                      mu3 = mu3p - 3 * mu * mu2 - mu**3
                      g1 = mu3 / np.power(mu2, 1.5)
                  out0 = default.copy()
                  place(out0, cond, g1)
                  output.append(out0)
  
              if 'k' in moments:
                  if g2 is None:
                      mu4p = self._munp(4, *goodargs)
                      if mu is None:
                          mu = self._munp(1, *goodargs)
                      if mu2 is None:
                          mu2p = self._munp(2, *goodargs)
                          mu2 = mu2p - mu * mu
                      if mu3 is None:
                          mu3p = self._munp(3, *goodargs)
                          mu3 = mu3p - 3 * mu * mu2 - mu**3
                      mu4 = mu4p - 4 * mu * mu3 - 6 * mu * mu * mu2 - mu**4
                      g2 = mu4 / mu2**2.0 - 3.0
                  out0 = default.copy()
                  place(out0, cond, g2)
                  output.append(out0)
          else:  # no valid args
              output = []
              for _ in moments:
                  out0 = default.copy()
                  output.append(out0)
  
          if len(output) == 1:
              return output[0]
          else:
              return tuple(output)
  
      def entropy(self, *args, **kwds):
          """
          Differential entropy of the RV.
  
          Parameters
          ----------
          arg1, arg2, arg3,... : array_like
              The shape parameter(s) for the distribution (see docstring of the
              instance object for more information).
          loc : array_like, optional
              Location parameter (default=0).
          scale : array_like, optional  (continuous distributions only).
              Scale parameter (default=1).
  
          Notes
          -----
          Entropy is defined base `e`:
  
          >>> drv = rv_discrete(values=((0, 1), (0.5, 0.5)))
          >>> np.allclose(drv.entropy(), np.log(2.0))
          True
  
          """
          args, loc, scale = self._parse_args(*args, **kwds)
          # NB: for discrete distributions scale=1 by construction in _parse_args
          args = tuple(map(asarray, args))
          cond0 = self._argcheck(*args) & (scale > 0) & (loc == loc)
          output = zeros(shape(cond0), 'd')
          place(output, (1-cond0), self.badvalue)
          goodargs = argsreduce(cond0, *args)
          # np.vectorize doesn't work when numargs == 0 in numpy 1.6.2.  Once the
          # lowest supported numpy version is >= 1.7.0, this special case can be
          # removed (see gh-4314).
          if self.numargs == 0:
              place(output, cond0, self._entropy() + log(scale))
          else:
              place(output, cond0, self.vecentropy(*goodargs) + log(scale))
          return output
  
      def moment(self, n, *args, **kwds):
          """
          n-th order non-central moment of distribution.
  
          Parameters
          ----------
          n : int, n >= 1
              Order of moment.
          arg1, arg2, arg3,... : float
              The shape parameter(s) for the distribution (see docstring of the
              instance object for more information).
          loc : array_like, optional
              location parameter (default=0)
          scale : array_like, optional
              scale parameter (default=1)
  
          """
          args, loc, scale = self._parse_args(*args, **kwds)
          if not (self._argcheck(*args) and (scale > 0)):
              return nan
          if (floor(n) != n):
              raise ValueError("Moment must be an integer.")
          if (n < 0):
              raise ValueError("Moment must be positive.")
          mu, mu2, g1, g2 = None, None, None, None
          if (n > 0) and (n < 5):
              if self._stats_has_moments:
                  mdict = {'moments': {1: 'm', 2: 'v', 3: 'vs', 4: 'vk'}[n]}
              else:
                  mdict = {}
              mu, mu2, g1, g2 = self._stats(*args, **mdict)
          val = _moment_from_stats(n, mu, mu2, g1, g2, self._munp, args)
  
          # Convert to transformed  X = L + S*Y
          # E[X^n] = E[(L+S*Y)^n] = L^n sum(comb(n, k)*(S/L)^k E[Y^k], k=0...n)
          if loc == 0:
              return scale**n * val
          else:
              result = 0
              fac = float(scale) / float(loc)
              for k in range(n):
                  valk = _moment_from_stats(k, mu, mu2, g1, g2, self._munp, args)
                  result += comb(n, k, exact=True)*(fac**k) * valk
              result += fac**n * val
              return result * loc**n
  
      def median(self, *args, **kwds):
          """
          Median of the distribution.
  
          Parameters
          ----------
          arg1, arg2, arg3,... : array_like
              The shape parameter(s) for the distribution (see docstring of the
              instance object for more information)
          loc : array_like, optional
              Location parameter, Default is 0.
          scale : array_like, optional
              Scale parameter, Default is 1.
  
          Returns
          -------
          median : float
              The median of the distribution.
  
          See Also
          --------
          stats.distributions.rv_discrete.ppf
              Inverse of the CDF
  
          """
          return self.ppf(0.5, *args, **kwds)
  
      def mean(self, *args, **kwds):
          """
          Mean of the distribution.
  
          Parameters
          ----------
          arg1, arg2, arg3,... : array_like
              The shape parameter(s) for the distribution (see docstring of the
              instance object for more information)
          loc : array_like, optional
              location parameter (default=0)
          scale : array_like, optional
              scale parameter (default=1)
  
          Returns
          -------
          mean : float
              the mean of the distribution
  
          """
          kwds['moments'] = 'm'
          res = self.stats(*args, **kwds)
          if isinstance(res, ndarray) and res.ndim == 0:
              return res[()]
          return res
  
      def var(self, *args, **kwds):
          """
          Variance of the distribution.
  
          Parameters
          ----------
          arg1, arg2, arg3,... : array_like
              The shape parameter(s) for the distribution (see docstring of the
              instance object for more information)
          loc : array_like, optional
              location parameter (default=0)
          scale : array_like, optional
              scale parameter (default=1)
  
          Returns
          -------
          var : float
              the variance of the distribution
  
          """
          kwds['moments'] = 'v'
          res = self.stats(*args, **kwds)
          if isinstance(res, ndarray) and res.ndim == 0:
              return res[()]
          return res
  
      def std(self, *args, **kwds):
          """
          Standard deviation of the distribution.
  
          Parameters
          ----------
          arg1, arg2, arg3,... : array_like
              The shape parameter(s) for the distribution (see docstring of the
              instance object for more information)
          loc : array_like, optional
              location parameter (default=0)
          scale : array_like, optional
              scale parameter (default=1)
  
          Returns
          -------
          std : float
              standard deviation of the distribution
  
          """
          kwds['moments'] = 'v'
          res = sqrt(self.stats(*args, **kwds))
          return res
  
      def interval(self, alpha, *args, **kwds):
          """
          Confidence interval with equal areas around the median.
  
          Parameters
          ----------
          alpha : array_like of float
              Probability that an rv will be drawn from the returned range.
              Each value should be in the range [0, 1].
          arg1, arg2, ... : array_like
              The shape parameter(s) for the distribution (see docstring of the
              instance object for more information).
          loc : array_like, optional
              location parameter, Default is 0.
          scale : array_like, optional
              scale parameter, Default is 1.
  
          Returns
          -------
          a, b : ndarray of float
              end-points of range that contain ``100 * alpha %`` of the rv's
              possible values.
  
          """
          alpha = asarray(alpha)
          if np.any((alpha > 1) | (alpha < 0)):
              raise ValueError("alpha must be between 0 and 1 inclusive")
          q1 = (1.0-alpha)/2
          q2 = (1.0+alpha)/2
          a = self.ppf(q1, *args, **kwds)
          b = self.ppf(q2, *args, **kwds)
          return a, b
  
  
  ##  continuous random variables: implement maybe later
  ##
  ##  hf  --- Hazard Function (PDF / SF)
  ##  chf  --- Cumulative hazard function (-log(SF))
  ##  psf --- Probability sparsity function (reciprocal of the pdf) in
  ##                units of percent-point-function (as a function of q).
  ##                Also, the derivative of the percent-point function.
  
  class rv_continuous(rv_generic):
      """
      A generic continuous random variable class meant for subclassing.
  
      `rv_continuous` is a base class to construct specific distribution classes
      and instances for continuous random variables. It cannot be used
      directly as a distribution.
  
      Parameters
      ----------
      momtype : int, optional
          The type of generic moment calculation to use: 0 for pdf, 1 (default)
          for ppf.
      a : float, optional
          Lower bound of the support of the distribution, default is minus
          infinity.
      b : float, optional
          Upper bound of the support of the distribution, default is plus
          infinity.
      xtol : float, optional
          The tolerance for fixed point calculation for generic ppf.
      badvalue : float, optional
          The value in a result arrays that indicates a value that for which
          some argument restriction is violated, default is np.nan.
      name : str, optional
          The name of the instance. This string is used to construct the default
          example for distributions.
      longname : str, optional
          This string is used as part of the first line of the docstring returned
          when a subclass has no docstring of its own. Note: `longname` exists
          for backwards compatibility, do not use for new subclasses.
      shapes : str, optional
          The shape of the distribution. For example ``"m, n"`` for a
          distribution that takes two integers as the two shape arguments for all
          its methods. If not provided, shape parameters will be inferred from
          the signature of the private methods, ``_pdf`` and ``_cdf`` of the
          instance.
      extradoc :  str, optional, deprecated
          This string is used as the last part of the docstring returned when a
          subclass has no docstring of its own. Note: `extradoc` exists for
          backwards compatibility, do not use for new subclasses.
      seed : None or int or ``numpy.random.RandomState`` instance, optional
          This parameter defines the RandomState object to use for drawing
          random variates.
          If None (or np.random), the global np.random state is used.
          If integer, it is used to seed the local RandomState instance.
          Default is None.
  
      Methods
      -------
      rvs
      pdf
      logpdf
      cdf
      logcdf
      sf
      logsf
      ppf
      isf
      moment
      stats
      entropy
      expect
      median
      mean
      std
      var
      interval
      __call__
      fit
      fit_loc_scale
      nnlf
  
      Notes
      -----
      Public methods of an instance of a distribution class (e.g., ``pdf``,
      ``cdf``) check their arguments and pass valid arguments to private,
      computational methods (``_pdf``, ``_cdf``). For ``pdf(x)``, ``x`` is valid
      if it is within the support of a distribution, ``self.a <= x <= self.b``.
      Whether a shape parameter is valid is decided by an ``_argcheck`` method
      (which defaults to checking that its arguments are strictly positive.)
  
      **Subclassing**
  
      New random variables can be defined by subclassing the `rv_continuous` class
      and re-defining at least the ``_pdf`` or the ``_cdf`` method (normalized
      to location 0 and scale 1).
  
      If positive argument checking is not correct for your RV
      then you will also need to re-define the ``_argcheck`` method.
  
      Correct, but potentially slow defaults exist for the remaining
      methods but for speed and/or accuracy you can over-ride::
  
        _logpdf, _cdf, _logcdf, _ppf, _rvs, _isf, _sf, _logsf
  
      Rarely would you override ``_isf``, ``_sf`` or ``_logsf``, but you could.
  
      **Methods that can be overwritten by subclasses**
      ::
  
        _rvs
        _pdf
        _cdf
        _sf
        _ppf
        _isf
        _stats
        _munp
        _entropy
        _argcheck
  
      There are additional (internal and private) generic methods that can
      be useful for cross-checking and for debugging, but might work in all
      cases when directly called.
  
      A note on ``shapes``: subclasses need not specify them explicitly. In this
      case, `shapes` will be automatically deduced from the signatures of the
      overridden methods (`pdf`, `cdf` etc).
      If, for some reason, you prefer to avoid relying on introspection, you can
      specify ``shapes`` explicitly as an argument to the instance constructor.
  
  
      **Frozen Distributions**
  
      Normally, you must provide shape parameters (and, optionally, location and
      scale parameters to each call of a method of a distribution.
  
      Alternatively, the object may be called (as a function) to fix the shape,
      location, and scale parameters returning a "frozen" continuous RV object:
  
      rv = generic(<shape(s)>, loc=0, scale=1)
          frozen RV object with the same methods but holding the given shape,
          location, and scale fixed
  
      **Statistics**
  
      Statistics are computed using numerical integration by default.
      For speed you can redefine this using ``_stats``:
  
       - take shape parameters and return mu, mu2, g1, g2
       - If you can't compute one of these, return it as None
       - Can also be defined with a keyword argument ``moments``, which is a
         string composed of "m", "v", "s", and/or "k".
         Only the components appearing in string should be computed and
         returned in the order "m", "v", "s", or "k"  with missing values
         returned as None.
  
      Alternatively, you can override ``_munp``, which takes ``n`` and shape
      parameters and returns the n-th non-central moment of the distribution.
  
      Examples
      --------
      To create a new Gaussian distribution, we would do the following:
  
      >>> from scipy.stats import rv_continuous
      >>> class gaussian_gen(rv_continuous):
      ...     "Gaussian distribution"
      ...     def _pdf(self, x):
      ...         return np.exp(-x**2 / 2.) / np.sqrt(2.0 * np.pi)
      >>> gaussian = gaussian_gen(name='gaussian')
  
      ``scipy.stats`` distributions are *instances*, so here we subclass
      `rv_continuous` and create an instance. With this, we now have
      a fully functional distribution with all relevant methods automagically
      generated by the framework.
  
      Note that above we defined a standard normal distribution, with zero mean
      and unit variance. Shifting and scaling of the distribution can be done
      by using ``loc`` and ``scale`` parameters: ``gaussian.pdf(x, loc, scale)``
      essentially computes ``y = (x - loc) / scale`` and
      ``gaussian._pdf(y) / scale``.
  
      """
      def __init__(self, momtype=1, a=None, b=None, xtol=1e-14,
                   badvalue=None, name=None, longname=None,
                   shapes=None, extradoc=None, seed=None):
  
          super(rv_continuous, self).__init__(seed)
  
          # save the ctor parameters, cf generic freeze
          self._ctor_param = dict(
              momtype=momtype, a=a, b=b, xtol=xtol,
              badvalue=badvalue, name=name, longname=longname,
              shapes=shapes, extradoc=extradoc, seed=seed)
  
          if badvalue is None:
              badvalue = nan
          if name is None:
              name = 'Distribution'
          self.badvalue = badvalue
          self.name = name
          self.a = a
          self.b = b
          if a is None:
              self.a = -inf
          if b is None:
              self.b = inf
          self.xtol = xtol
          self._size = 1
          self.moment_type = momtype
          self.shapes = shapes
          self._construct_argparser(meths_to_inspect=[self._pdf, self._cdf],
                                    locscale_in='loc=0, scale=1',
                                    locscale_out='loc, scale')
  
          # nin correction
          self._ppfvec = vectorize(self._ppf_single, otypes='d')
          self._ppfvec.nin = self.numargs + 1
          self.vecentropy = vectorize(self._entropy, otypes='d')
          self._cdfvec = vectorize(self._cdf_single, otypes='d')
          self._cdfvec.nin = self.numargs + 1
  
          self.extradoc = extradoc
          if momtype == 0:
              self.generic_moment = vectorize(self._mom0_sc, otypes='d')
          else:
              self.generic_moment = vectorize(self._mom1_sc, otypes='d')
          # Because of the *args argument of _mom0_sc, vectorize cannot count the
          # number of arguments correctly.
          self.generic_moment.nin = self.numargs + 1
  
          if longname is None:
              if name[0] in ['aeiouAEIOU']:
                  hstr = "An "
              else:
                  hstr = "A "
              longname = hstr + name
  
          if sys.flags.optimize < 2:
              # Skip adding docstrings if interpreter is run with -OO
              if self.__doc__ is None:
                  self._construct_default_doc(longname=longname,
                                              extradoc=extradoc,
                                              docdict=docdict,
                                              discrete='continuous')
              else:
                  dct = dict(distcont)
                  self._construct_doc(docdict, dct.get(self.name))
  
      def _updated_ctor_param(self):
          """ Return the current version of _ctor_param, possibly updated by user.
  
              Used by freezing and pickling.
              Keep this in sync with the signature of __init__.
          """
          dct = self._ctor_param.copy()
          dct['a'] = self.a
          dct['b'] = self.b
          dct['xtol'] = self.xtol
          dct['badvalue'] = self.badvalue
          dct['name'] = self.name
          dct['shapes'] = self.shapes
          dct['extradoc'] = self.extradoc
          return dct
  
      def _ppf_to_solve(self, x, q, *args):
          return self.cdf(*(x, )+args)-q
  
      def _ppf_single(self, q, *args):
          left = right = None
          if self.a > -np.inf:
              left = self.a
          if self.b < np.inf:
              right = self.b
  
          factor = 10.
          if not left:  # i.e. self.a = -inf
              left = -1.*factor
              while self._ppf_to_solve(left, q, *args) > 0.:
                  right = left
                  left *= factor
              # left is now such that cdf(left) < q
          if not right:  # i.e. self.b = inf
              right = factor
              while self._ppf_to_solve(right, q, *args) < 0.:
                  left = right
                  right *= factor
              # right is now such that cdf(right) > q
  
          return optimize.brentq(self._ppf_to_solve,
                                 left, right, args=(q,)+args, xtol=self.xtol)
  
      # moment from definition
      def _mom_integ0(self, x, m, *args):
          return x**m * self.pdf(x, *args)
  
      def _mom0_sc(self, m, *args):
          return integrate.quad(self._mom_integ0, self.a, self.b,
                                args=(m,)+args)[0]
  
      # moment calculated using ppf
      def _mom_integ1(self, q, m, *args):
          return (self.ppf(q, *args))**m
  
      def _mom1_sc(self, m, *args):
          return integrate.quad(self._mom_integ1, 0, 1, args=(m,)+args)[0]
  
      def _pdf(self, x, *args):
          return derivative(self._cdf, x, dx=1e-5, args=args, order=5)
  
      ## Could also define any of these
      def _logpdf(self, x, *args):
          return log(self._pdf(x, *args))
  
      def _cdf_single(self, x, *args):
          return integrate.quad(self._pdf, self.a, x, args=args)[0]
  
      def _cdf(self, x, *args):
          return self._cdfvec(x, *args)
  
      ## generic _argcheck, _logcdf, _sf, _logsf, _ppf, _isf, _rvs are defined
      ## in rv_generic
  
      def pdf(self, x, *args, **kwds):
          """
          Probability density function at x of the given RV.
  
          Parameters
          ----------
          x : array_like
              quantiles
          arg1, arg2, arg3,... : array_like
              The shape parameter(s) for the distribution (see docstring of the
              instance object for more information)
          loc : array_like, optional
              location parameter (default=0)
          scale : array_like, optional
              scale parameter (default=1)
  
          Returns
          -------
          pdf : ndarray
              Probability density function evaluated at x
  
          """
          args, loc, scale = self._parse_args(*args, **kwds)
          x, loc, scale = map(asarray, (x, loc, scale))
          args = tuple(map(asarray, args))
          dtyp = np.find_common_type([x.dtype, np.float64], [])
          x = np.asarray((x - loc)/scale, dtype=dtyp)
          cond0 = self._argcheck(*args) & (scale > 0)
          cond1 = (scale > 0) & (x >= self.a) & (x <= self.b)
          cond = cond0 & cond1
          output = zeros(shape(cond), dtyp)
          putmask(output, (1-cond0)+np.isnan(x), self.badvalue)
          if np.any(cond):
              goodargs = argsreduce(cond, *((x,)+args+(scale,)))
              scale, goodargs = goodargs[-1], goodargs[:-1]
              place(output, cond, self._pdf(*goodargs) / scale)
          if output.ndim == 0:
              return output[()]
          return output
  
      def logpdf(self, x, *args, **kwds):
          """
          Log of the probability density function at x of the given RV.
  
          This uses a more numerically accurate calculation if available.
  
          Parameters
          ----------
          x : array_like
              quantiles
          arg1, arg2, arg3,... : array_like
              The shape parameter(s) for the distribution (see docstring of the
              instance object for more information)
          loc : array_like, optional
              location parameter (default=0)
          scale : array_like, optional
              scale parameter (default=1)
  
          Returns
          -------
          logpdf : array_like
              Log of the probability density function evaluated at x
  
          """
          args, loc, scale = self._parse_args(*args, **kwds)
          x, loc, scale = map(asarray, (x, loc, scale))
          args = tuple(map(asarray, args))
          dtyp = np.find_common_type([x.dtype, np.float64], [])
          x = np.asarray((x - loc)/scale, dtype=dtyp)
          cond0 = self._argcheck(*args) & (scale > 0)
          cond1 = (scale > 0) & (x >= self.a) & (x <= self.b)
          cond = cond0 & cond1
          output = empty(shape(cond), dtyp)
          output.fill(NINF)
          putmask(output, (1-cond0)+np.isnan(x), self.badvalue)
          if np.any(cond):
              goodargs = argsreduce(cond, *((x,)+args+(scale,)))
              scale, goodargs = goodargs[-1], goodargs[:-1]
              place(output, cond, self._logpdf(*goodargs) - log(scale))
          if output.ndim == 0:
              return output[()]
          return output
  
      def cdf(self, x, *args, **kwds):
          """
          Cumulative distribution function of the given RV.
  
          Parameters
          ----------
          x : array_like
              quantiles
          arg1, arg2, arg3,... : array_like
              The shape parameter(s) for the distribution (see docstring of the
              instance object for more information)
          loc : array_like, optional
              location parameter (default=0)
          scale : array_like, optional
              scale parameter (default=1)
  
          Returns
          -------
          cdf : ndarray
              Cumulative distribution function evaluated at `x`
  
          """
          args, loc, scale = self._parse_args(*args, **kwds)
          x, loc, scale = map(asarray, (x, loc, scale))
          args = tuple(map(asarray, args))
          dtyp = np.find_common_type([x.dtype, np.float64], [])
          x = np.asarray((x - loc)/scale, dtype=dtyp)
          cond0 = self._argcheck(*args) & (scale > 0)
          cond1 = (scale > 0) & (x > self.a) & (x < self.b)
          cond2 = (x >= self.b) & cond0
          cond = cond0 & cond1
          output = zeros(shape(cond), dtyp)
          place(output, (1-cond0)+np.isnan(x), self.badvalue)
          place(output, cond2, 1.0)
          if np.any(cond):  # call only if at least 1 entry
              goodargs = argsreduce(cond, *((x,)+args))
              place(output, cond, self._cdf(*goodargs))
          if output.ndim == 0:
              return output[()]
          return output
  
      def logcdf(self, x, *args, **kwds):
          """
          Log of the cumulative distribution function at x of the given RV.
  
          Parameters
          ----------
          x : array_like
              quantiles
          arg1, arg2, arg3,... : array_like
              The shape parameter(s) for the distribution (see docstring of the
              instance object for more information)
          loc : array_like, optional
              location parameter (default=0)
          scale : array_like, optional
              scale parameter (default=1)
  
          Returns
          -------
          logcdf : array_like
              Log of the cumulative distribution function evaluated at x
  
          """
          args, loc, scale = self._parse_args(*args, **kwds)
          x, loc, scale = map(asarray, (x, loc, scale))
          args = tuple(map(asarray, args))
          dtyp = np.find_common_type([x.dtype, np.float64], [])
          x = np.asarray((x - loc)/scale, dtype=dtyp)
          cond0 = self._argcheck(*args) & (scale > 0)
          cond1 = (scale > 0) & (x > self.a) & (x < self.b)
          cond2 = (x >= self.b) & cond0
          cond = cond0 & cond1
          output = empty(shape(cond), dtyp)
          output.fill(NINF)
          place(output, (1-cond0)*(cond1 == cond1)+np.isnan(x), self.badvalue)
          place(output, cond2, 0.0)
          if np.any(cond):  # call only if at least 1 entry
              goodargs = argsreduce(cond, *((x,)+args))
              place(output, cond, self._logcdf(*goodargs))
          if output.ndim == 0:
              return output[()]
          return output
  
      def sf(self, x, *args, **kwds):
          """
          Survival function (1 - `cdf`) at x of the given RV.
  
          Parameters
          ----------
          x : array_like
              quantiles
          arg1, arg2, arg3,... : array_like
              The shape parameter(s) for the distribution (see docstring of the
              instance object for more information)
          loc : array_like, optional
              location parameter (default=0)
          scale : array_like, optional
              scale parameter (default=1)
  
          Returns
          -------
          sf : array_like
              Survival function evaluated at x
  
          """
          args, loc, scale = self._parse_args(*args, **kwds)
          x, loc, scale = map(asarray, (x, loc, scale))
          args = tuple(map(asarray, args))
          dtyp = np.find_common_type([x.dtype, np.float64], [])
          x = np.asarray((x - loc)/scale, dtype=dtyp)
          cond0 = self._argcheck(*args) & (scale > 0)
          cond1 = (scale > 0) & (x > self.a) & (x < self.b)
          cond2 = cond0 & (x <= self.a)
          cond = cond0 & cond1
          output = zeros(shape(cond), dtyp)
          place(output, (1-cond0)+np.isnan(x), self.badvalue)
          place(output, cond2, 1.0)
          if np.any(cond):
              goodargs = argsreduce(cond, *((x,)+args))
              place(output, cond, self._sf(*goodargs))
          if output.ndim == 0:
              return output[()]
          return output
  
      def logsf(self, x, *args, **kwds):
          """
          Log of the survival function of the given RV.
  
          Returns the log of the "survival function," defined as (1 - `cdf`),
          evaluated at `x`.
  
          Parameters
          ----------
          x : array_like
              quantiles
          arg1, arg2, arg3,... : array_like
              The shape parameter(s) for the distribution (see docstring of the
              instance object for more information)
          loc : array_like, optional
              location parameter (default=0)
          scale : array_like, optional
              scale parameter (default=1)
  
          Returns
          -------
          logsf : ndarray
              Log of the survival function evaluated at `x`.
  
          """
          args, loc, scale = self._parse_args(*args, **kwds)
          x, loc, scale = map(asarray, (x, loc, scale))
          args = tuple(map(asarray, args))
          dtyp = np.find_common_type([x.dtype, np.float64], [])
          x = np.asarray((x - loc)/scale, dtype=dtyp)
          cond0 = self._argcheck(*args) & (scale > 0)
          cond1 = (scale > 0) & (x > self.a) & (x < self.b)
          cond2 = cond0 & (x <= self.a)
          cond = cond0 & cond1
          output = empty(shape(cond), dtyp)
          output.fill(NINF)
          place(output, (1-cond0)+np.isnan(x), self.badvalue)
          place(output, cond2, 0.0)
          if np.any(cond):
              goodargs = argsreduce(cond, *((x,)+args))
              place(output, cond, self._logsf(*goodargs))
          if output.ndim == 0:
              return output[()]
          return output
  
      def ppf(self, q, *args, **kwds):
          """
          Percent point function (inverse of `cdf`) at q of the given RV.
  
          Parameters
          ----------
          q : array_like
              lower tail probability
          arg1, arg2, arg3,... : array_like
              The shape parameter(s) for the distribution (see docstring of the
              instance object for more information)
          loc : array_like, optional
              location parameter (default=0)
          scale : array_like, optional
              scale parameter (default=1)
  
          Returns
          -------
          x : array_like
              quantile corresponding to the lower tail probability q.
  
          """
          args, loc, scale = self._parse_args(*args, **kwds)
          q, loc, scale = map(asarray, (q, loc, scale))
          args = tuple(map(asarray, args))
          cond0 = self._argcheck(*args) & (scale > 0) & (loc == loc)
          cond1 = (0 < q) & (q < 1)
          cond2 = cond0 & (q == 0)
          cond3 = cond0 & (q == 1)
          cond = cond0 & cond1
          output = valarray(shape(cond), value=self.badvalue)
  
          lower_bound = self.a * scale + loc
          upper_bound = self.b * scale + loc
          place(output, cond2, argsreduce(cond2, lower_bound)[0])
          place(output, cond3, argsreduce(cond3, upper_bound)[0])
  
          if np.any(cond):  # call only if at least 1 entry
              goodargs = argsreduce(cond, *((q,)+args+(scale, loc)))
              scale, loc, goodargs = goodargs[-2], goodargs[-1], goodargs[:-2]
              place(output, cond, self._ppf(*goodargs) * scale + loc)
          if output.ndim == 0:
              return output[()]
          return output
  
      def isf(self, q, *args, **kwds):
          """
          Inverse survival function (inverse of `sf`) at q of the given RV.
  
          Parameters
          ----------
          q : array_like
              upper tail probability
          arg1, arg2, arg3,... : array_like
              The shape parameter(s) for the distribution (see docstring of the
              instance object for more information)
          loc : array_like, optional
              location parameter (default=0)
          scale : array_like, optional
              scale parameter (default=1)
  
          Returns
          -------
          x : ndarray or scalar
              Quantile corresponding to the upper tail probability q.
  
          """
          args, loc, scale = self._parse_args(*args, **kwds)
          q, loc, scale = map(asarray, (q, loc, scale))
          args = tuple(map(asarray, args))
          cond0 = self._argcheck(*args) & (scale > 0) & (loc == loc)
          cond1 = (0 < q) & (q < 1)
          cond2 = cond0 & (q == 1)
          cond3 = cond0 & (q == 0)
          cond = cond0 & cond1
          output = valarray(shape(cond), value=self.badvalue)
  
          lower_bound = self.a * scale + loc
          upper_bound = self.b * scale + loc
          place(output, cond2, argsreduce(cond2, lower_bound)[0])
          place(output, cond3, argsreduce(cond3, upper_bound)[0])
  
          if np.any(cond):
              goodargs = argsreduce(cond, *((q,)+args+(scale, loc)))
              scale, loc, goodargs = goodargs[-2], goodargs[-1], goodargs[:-2]
              place(output, cond, self._isf(*goodargs) * scale + loc)
          if output.ndim == 0:
              return output[()]
          return output
  
      def _nnlf(self, x, *args):
          return -np.sum(self._logpdf(x, *args), axis=0)
  
      def nnlf(self, theta, x):
          '''Return negative loglikelihood function.
  
          Notes
          -----
          This is ``-sum(log pdf(x, theta), axis=0)`` where `theta` are the
          parameters (including loc and scale).
          '''
          try:
              loc = theta[-2]
              scale = theta[-1]
              args = tuple(theta[:-2])
          except IndexError:
              raise ValueError("Not enough input arguments.")
          if not self._argcheck(*args) or scale <= 0:
              return inf
          x = asarray((x-loc) / scale)
          cond0 = (x <= self.a) | (self.b <= x)
          if np.any(cond0):
              return inf
          else:
              N = len(x)
              return self._nnlf(x, *args) + N * log(scale)
  
      def _penalized_nnlf(self, theta, x):
          ''' Return negative loglikelihood function,
          i.e., - sum (log pdf(x, theta), axis=0)
             where theta are the parameters (including loc and scale)
          '''
          try:
              loc = theta[-2]
              scale = theta[-1]
              args = tuple(theta[:-2])
          except IndexError:
              raise ValueError("Not enough input arguments.")
          if not self._argcheck(*args) or scale <= 0:
              return inf
          x = asarray((x-loc) / scale)
  
          loginf = log(_XMAX)
  
          if np.isneginf(self.a).all() and np.isinf(self.b).all():
              Nbad = 0
          else:
              cond0 = (x <= self.a) | (self.b <= x)
              Nbad = np.sum(cond0)
              if Nbad > 0:
                  x = argsreduce(~cond0, x)[0]
  
          N = len(x)
          return self._nnlf(x, *args) + N*log(scale) + Nbad * 100.0 * loginf
  
      # return starting point for fit (shape arguments + loc + scale)
      def _fitstart(self, data, args=None):
          if args is None:
              args = (1.0,)*self.numargs
          loc, scale = self._fit_loc_scale_support(data, *args)
          return args + (loc, scale)
  
      # Return the (possibly reduced) function to optimize in order to find MLE
      #  estimates for the .fit method
      def _reduce_func(self, args, kwds):
          # First of all, convert fshapes params to fnum: eg for stats.beta,
          # shapes='a, b'. To fix `a`, can specify either `f1` or `fa`.
          # Convert the latter into the former.
          if self.shapes:
              shapes = self.shapes.replace(',', ' ').split()
              for j, s in enumerate(shapes):
                  val = kwds.pop('f' + s, None) or kwds.pop('fix_' + s, None)
                  if val is not None:
                      key = 'f%d' % j
                      if key in kwds:
                          raise ValueError("Duplicate entry for %s." % key)
                      else:
                          kwds[key] = val
  
          args = list(args)
          Nargs = len(args)
          fixedn = []
          names = ['f%d' % n for n in range(Nargs - 2)] + ['floc', 'fscale']
          x0 = []
          for n, key in enumerate(names):
              if key in kwds:
                  fixedn.append(n)
                  args[n] = kwds.pop(key)
              else:
                  x0.append(args[n])
  
          if len(fixedn) == 0:
              func = self._penalized_nnlf
              restore = None
          else:
              if len(fixedn) == Nargs:
                  raise ValueError(
                      "All parameters fixed. There is nothing to optimize.")
  
              def restore(args, theta):
                  # Replace with theta for all numbers not in fixedn
                  # This allows the non-fixed values to vary, but
                  #  we still call self.nnlf with all parameters.
                  i = 0
                  for n in range(Nargs):
                      if n not in fixedn:
                          args[n] = theta[i]
                          i += 1
                  return args
  
              def func(theta, x):
                  newtheta = restore(args[:], theta)
                  return self._penalized_nnlf(newtheta, x)
  
          return x0, func, restore, args
  
      def fit(self, data, *args, **kwds):
          """
          Return MLEs for shape, location, and scale parameters from data.
  
          MLE stands for Maximum Likelihood Estimate.  Starting estimates for
          the fit are given by input arguments; for any arguments not provided
          with starting estimates, ``self._fitstart(data)`` is called to generate
          such.
  
          One can hold some parameters fixed to specific values by passing in
          keyword arguments ``f0``, ``f1``, ..., ``fn`` (for shape parameters)
          and ``floc`` and ``fscale`` (for location and scale parameters,
          respectively).
  
          Parameters
          ----------
          data : array_like
              Data to use in calculating the MLEs.
          args : floats, optional
              Starting value(s) for any shape-characterizing arguments (those not
              provided will be determined by a call to ``_fitstart(data)``).
              No default value.
          kwds : floats, optional
              Starting values for the location and scale parameters; no default.
              Special keyword arguments are recognized as holding certain
              parameters fixed:
  
              - f0...fn : hold respective shape parameters fixed.
                Alternatively, shape parameters to fix can be specified by name.
                For example, if ``self.shapes == "a, b"``, ``fa``and ``fix_a``
                are equivalent to ``f0``, and ``fb`` and ``fix_b`` are
                equivalent to ``f1``.
  
              - floc : hold location parameter fixed to specified value.
  
              - fscale : hold scale parameter fixed to specified value.
  
              - optimizer : The optimizer to use.  The optimizer must take ``func``,
                and starting position as the first two arguments,
                plus ``args`` (for extra arguments to pass to the
                function to be optimized) and ``disp=0`` to suppress
                output as keyword arguments.
  
          Returns
          -------
          shape, loc, scale : tuple of floats
              MLEs for any shape statistics, followed by those for location and
              scale.
  
          Notes
          -----
          This fit is computed by maximizing a log-likelihood function, with
          penalty applied for samples outside of range of the distribution. The
          returned answer is not guaranteed to be the globally optimal MLE, it
          may only be locally optimal, or the optimization may fail altogether.
  
  
          Examples
          --------
  
          Generate some data to fit: draw random variates from the `beta`
          distribution
  
          >>> from scipy.stats import beta
          >>> a, b = 1., 2.
          >>> x = beta.rvs(a, b, size=1000)
  
          Now we can fit all four parameters (``a``, ``b``, ``loc`` and ``scale``):
  
          >>> a1, b1, loc1, scale1 = beta.fit(x)
  
          We can also use some prior knowledge about the dataset: let's keep
          ``loc`` and ``scale`` fixed:
  
          >>> a1, b1, loc1, scale1 = beta.fit(x, floc=0, fscale=1)
          >>> loc1, scale1
          (0, 1)
  
          We can also keep shape parameters fixed by using ``f``-keywords. To
          keep the zero-th shape parameter ``a`` equal 1, use ``f0=1`` or,
          equivalently, ``fa=1``:
  
          >>> a1, b1, loc1, scale1 = beta.fit(x, fa=1, floc=0, fscale=1)
          >>> a1
          1
  
          """
          Narg = len(args)
          if Narg > self.numargs:
              raise TypeError("Too many input arguments.")
  
          start = [None]*2
          if (Narg < self.numargs) or not ('loc' in kwds and
                                           'scale' in kwds):
              # get distribution specific starting locations
              start = self._fitstart(data)
              args += start[Narg:-2]
          loc = kwds.pop('loc', start[-2])
          scale = kwds.pop('scale', start[-1])
          args += (loc, scale)
          x0, func, restore, args = self._reduce_func(args, kwds)
  
          optimizer = kwds.pop('optimizer', optimize.fmin)
          # convert string to function in scipy.optimize
          if not callable(optimizer) and isinstance(optimizer, string_types):
              if not optimizer.startswith('fmin_'):
                  optimizer = "fmin_"+optimizer
              if optimizer == 'fmin_':
                  optimizer = 'fmin'
              try:
                  optimizer = getattr(optimize, optimizer)
              except AttributeError:
                  raise ValueError("%s is not a valid optimizer" % optimizer)
  
          # by now kwds must be empty, since everybody took what they needed
          if kwds:
              raise TypeError("Unknown arguments: %s." % kwds)
  
          vals = optimizer(func, x0, args=(ravel(data),), disp=0)
          if restore is not None:
              vals = restore(args, vals)
          vals = tuple(vals)
          return vals
  
      def _fit_loc_scale_support(self, data, *args):
          """
          Estimate loc and scale parameters from data accounting for support.
  
          Parameters
          ----------
          data : array_like
              Data to fit.
          arg1, arg2, arg3,... : array_like
              The shape parameter(s) for the distribution (see docstring of the
              instance object for more information).
  
          Returns
          -------
          Lhat : float
              Estimated location parameter for the data.
          Shat : float
              Estimated scale parameter for the data.
  
          """
          data = np.asarray(data)
  
          # Estimate location and scale according to the method of moments.
          loc_hat, scale_hat = self.fit_loc_scale(data, *args)
  
          # Compute the support according to the shape parameters.
          self._argcheck(*args)
          a, b = self.a, self.b
          support_width = b - a
  
          # If the support is empty then return the moment-based estimates.
          if support_width <= 0:
              return loc_hat, scale_hat
  
          # Compute the proposed support according to the loc and scale estimates.
          a_hat = loc_hat + a * scale_hat
          b_hat = loc_hat + b * scale_hat
  
          # Use the moment-based estimates if they are compatible with the data.
          data_a = np.min(data)
          data_b = np.max(data)
          if a_hat < data_a and data_b < b_hat:
              return loc_hat, scale_hat
  
          # Otherwise find other estimates that are compatible with the data.
          data_width = data_b - data_a
          rel_margin = 0.1
          margin = data_width * rel_margin
  
          # For a finite interval, both the location and scale
          # should have interesting values.
          if support_width < np.inf:
              loc_hat = (data_a - a) - margin
              scale_hat = (data_width + 2 * margin) / support_width
              return loc_hat, scale_hat
  
          # For a one-sided interval, use only an interesting location parameter.
          if a > -np.inf:
              return (data_a - a) - margin, 1
          elif b < np.inf:
              return (data_b - b) + margin, 1
          else:
              raise RuntimeError
  
      def fit_loc_scale(self, data, *args):
          """
          Estimate loc and scale parameters from data using 1st and 2nd moments.
  
          Parameters
          ----------
          data : array_like
              Data to fit.
          arg1, arg2, arg3,... : array_like
              The shape parameter(s) for the distribution (see docstring of the
              instance object for more information).
  
          Returns
          -------
          Lhat : float
              Estimated location parameter for the data.
          Shat : float
              Estimated scale parameter for the data.
  
          """
          mu, mu2 = self.stats(*args, **{'moments': 'mv'})
          tmp = asarray(data)
          muhat = tmp.mean()
          mu2hat = tmp.var()
          Shat = sqrt(mu2hat / mu2)
          Lhat = muhat - Shat*mu
          if not np.isfinite(Lhat):
              Lhat = 0
          if not (np.isfinite(Shat) and (0 < Shat)):
              Shat = 1
          return Lhat, Shat
  
      def _entropy(self, *args):
          def integ(x):
              val = self._pdf(x, *args)
              return entr(val)
  
          # upper limit is often inf, so suppress warnings when integrating
          olderr = np.seterr(over='ignore')
          h = integrate.quad(integ, self.a, self.b)[0]
          np.seterr(**olderr)
  
          if not np.isnan(h):
              return h
          else:
              # try with different limits if integration problems
              low, upp = self.ppf([1e-10, 1. - 1e-10], *args)
              if np.isinf(self.b):
                  upper = upp
              else:
                  upper = self.b
              if np.isinf(self.a):
                  lower = low
              else:
                  lower = self.a
              return integrate.quad(integ, lower, upper)[0]
  
      def expect(self, func=None, args=(), loc=0, scale=1, lb=None, ub=None,
                 conditional=False, **kwds):
          """Calculate expected value of a function with respect to the
          distribution.
  
          The expected value of a function ``f(x)`` with respect to a
          distribution ``dist`` is defined as::
  
                      ubound
              E[x] = Integral(f(x) * dist.pdf(x))
                      lbound
  
          Parameters
          ----------
          func : callable, optional
              Function for which integral is calculated. Takes only one argument.
              The default is the identity mapping f(x) = x.
          args : tuple, optional
              Shape parameters of the distribution.
          loc : float, optional
              Location parameter (default=0).
          scale : float, optional
              Scale parameter (default=1).
          lb, ub : scalar, optional
              Lower and upper bound for integration. Default is set to the
              support of the distribution.
          conditional : bool, optional
              If True, the integral is corrected by the conditional probability
              of the integration interval.  The return value is the expectation
              of the function, conditional on being in the given interval.
              Default is False.
  
          Additional keyword arguments are passed to the integration routine.
  
          Returns
          -------
          expect : float
              The calculated expected value.
  
          Notes
          -----
          The integration behavior of this function is inherited from
          `integrate.quad`.
  
          """
          lockwds = {'loc': loc,
                     'scale': scale}
          self._argcheck(*args)
          if func is None:
              def fun(x, *args):
                  return x * self.pdf(x, *args, **lockwds)
          else:
              def fun(x, *args):
                  return func(x) * self.pdf(x, *args, **lockwds)
          if lb is None:
              lb = loc + self.a * scale
          if ub is None:
              ub = loc + self.b * scale
          if conditional:
              invfac = (self.sf(lb, *args, **lockwds)
                        - self.sf(ub, *args, **lockwds))
          else:
              invfac = 1.0
          kwds['args'] = args
          # Silence floating point warnings from integration.
          olderr = np.seterr(all='ignore')
          vals = integrate.quad(fun, lb, ub, **kwds)[0] / invfac
          np.seterr(**olderr)
          return vals
  
  
  ## Handlers for generic case where xk and pk are given
  ## The _drv prefix probably means discrete random variable.
  
  def _drv_pmf(self, xk, *args):
      try:
          return self.P[xk]
      except KeyError:
          return 0.0
  
  
  def _drv_cdf(self, xk, *args):
      indx = argmax((self.xk > xk), axis=-1)-1
      return self.F[self.xk[indx]]
  
  
  def _drv_ppf(self, q, *args):
      indx = argmax((self.qvals >= q), axis=-1)
      return self.Finv[self.qvals[indx]]
  
  
  def _drv_nonzero(self, k, *args):
      return 1
  
  
  def _drv_moment(self, n, *args):
      n = asarray(n)
      return np.sum(self.xk**n[np.newaxis, ...] * self.pk, axis=0)
  
  
  def _drv_moment_gen(self, t, *args):
      t = asarray(t)
      return np.sum(exp(self.xk * t[np.newaxis, ...]) * self.pk, axis=0)
  
  
  def _drv2_moment(self, n, *args):
      """Non-central moment of discrete distribution."""
      def fun(x):
          return np.power(x, n) * self._pmf(x, *args)
      return _expect(fun, self.a, self.b, self.ppf(0.5, *args), self.inc)
  
  
  def _drv2_ppfsingle(self, q, *args):  # Use basic bisection algorithm
      b = self.b
      a = self.a
      if isinf(b):            # Be sure ending point is > q
          b = int(max(100*q, 10))
          while 1:
              if b >= self.b:
                  qb = 1.0
                  break
              qb = self._cdf(b, *args)
              if (qb < q):
                  b += 10
              else:
                  break
      else:
          qb = 1.0
      if isinf(a):    # be sure starting point < q
          a = int(min(-100*q, -10))
          while 1:
              if a <= self.a:
                  qb = 0.0
                  break
              qa = self._cdf(a, *args)
              if (qa > q):
                  a -= 10
              else:
                  break
      else:
          qa = self._cdf(a, *args)
  
      while 1:
          if (qa == q):
              return a
          if (qb == q):
              return b
          if b <= a+1:
              # testcase: return wrong number at lower index
              # python -c "from scipy.stats import zipf;print zipf.ppf(0.01, 2)" wrong
              # python -c "from scipy.stats import zipf;print zipf.ppf([0.01, 0.61, 0.77, 0.83], 2)"
              # python -c "from scipy.stats import logser;print logser.ppf([0.1, 0.66, 0.86, 0.93], 0.6)"
              if qa > q:
                  return a
              else:
                  return b
          c = int((a+b)/2.0)
          qc = self._cdf(c, *args)
          if (qc < q):
              if a != c:
                  a = c
              else:
                  raise RuntimeError('updating stopped, endless loop')
              qa = qc
          elif (qc > q):
              if b != c:
                  b = c
              else:
                  raise RuntimeError('updating stopped, endless loop')
              qb = qc
          else:
              return c
  
  
  def entropy(pk, qk=None, base=None):
      """Calculate the entropy of a distribution for given probability values.
  
      If only probabilities `pk` are given, the entropy is calculated as
      ``S = -sum(pk * log(pk), axis=0)``.
  
      If `qk` is not None, then compute the Kullback-Leibler divergence
      ``S = sum(pk * log(pk / qk), axis=0)``.
  
      This routine will normalize `pk` and `qk` if they don't sum to 1.
  
      Parameters
      ----------
      pk : sequence
          Defines the (discrete) distribution. ``pk[i]`` is the (possibly
          unnormalized) probability of event ``i``.
      qk : sequence, optional
          Sequence against which the relative entropy is computed. Should be in
          the same format as `pk`.
      base : float, optional
          The logarithmic base to use, defaults to ``e`` (natural logarithm).
  
      Returns
      -------
      S : float
          The calculated entropy.
  
      """
      pk = asarray(pk)
      pk = 1.0*pk / np.sum(pk, axis=0)
      if qk is None:
          vec = entr(pk)
      else:
          qk = asarray(qk)
          if len(qk) != len(pk):
              raise ValueError("qk and pk must have same length.")
          qk = 1.0*qk / np.sum(qk, axis=0)
          vec = kl_div(pk, qk)
      S = np.sum(vec, axis=0)
      if base is not None:
          S /= log(base)
      return S
  
  
  # Must over-ride one of _pmf or _cdf or pass in
  #  x_k, p(x_k) lists in initialization
  
  class rv_discrete(rv_generic):
      """
      A generic discrete random variable class meant for subclassing.
  
      `rv_discrete` is a base class to construct specific distribution classes
      and instances for discrete random variables. It can also be used
      to construct an arbitrary distribution defined by a list of support
      points and corresponding probabilities.
  
      Parameters
      ----------
      a : float, optional
          Lower bound of the support of the distribution, default: 0
      b : float, optional
          Upper bound of the support of the distribution, default: plus infinity
      moment_tol : float, optional
          The tolerance for the generic calculation of moments.
      values : tuple of two array_like, optional
          ``(xk, pk)`` where ``xk`` are integers with non-zero
          probabilities ``pk``  with ``sum(pk) = 1``.
      inc : integer, optional
          Increment for the support of the distribution.
          Default is 1. (other values have not been tested)
      badvalue : float, optional
          The value in a result arrays that indicates a value that for which
          some argument restriction is violated, default is np.nan.
      name : str, optional
          The name of the instance. This string is used to construct the default
          example for distributions.
      longname : str, optional
          This string is used as part of the first line of the docstring returned
          when a subclass has no docstring of its own. Note: `longname` exists
          for backwards compatibility, do not use for new subclasses.
      shapes : str, optional
          The shape of the distribution. For example "m, n" for a distribution
          that takes two integers as the two shape arguments for all its methods
          If not provided, shape parameters will be inferred from
          the signatures of the private methods, ``_pmf`` and ``_cdf`` of
          the instance.
      extradoc :  str, optional
          This string is used as the last part of the docstring returned when a
          subclass has no docstring of its own. Note: `extradoc` exists for
          backwards compatibility, do not use for new subclasses.
      seed : None or int or ``numpy.random.RandomState`` instance, optional
          This parameter defines the RandomState object to use for drawing
          random variates.
          If None, the global np.random state is used.
          If integer, it is used to seed the local RandomState instance.
          Default is None.
  
      Methods
      -------
      rvs
      pmf
      logpmf
      cdf
      logcdf
      sf
      logsf
      ppf
      isf
      moment
      stats
      entropy
      expect
      median
      mean
      std
      var
      interval
      __call__
  
  
      Notes
      -----
  
      This class is similar to `rv_continuous`, the main differences being:
  
      - the support of the distribution is a set of integers
      - instead of the probability density function, ``pdf`` (and the
        corresponding private ``_pdf``), this class defines the
        *probability mass function*, `pmf` (and the corresponding
        private ``_pmf``.)
      - scale parameter is not defined.
  
      To create a new discrete distribution, we would do the following:
  
      >>> from scipy.stats import rv_discrete
      >>> class poisson_gen(rv_discrete):
      ...     "Poisson distribution"
      ...     def _pmf(self, k, mu):
      ...         return exp(-mu) * mu**k / factorial(k)
  
      and create an instance::
  
      >>> poisson = poisson_gen(name="poisson")
  
      Note that above we defined the Poisson distribution in the standard form.
      Shifting the distribution can be done by providing the ``loc`` parameter
      to the methods of the instance. For example, ``poisson.pmf(x, mu, loc)``
      delegates the work to ``poisson._pmf(x-loc, mu)``.
  
      **Discrete distributions from a list of probabilities**
  
      Alternatively, you can construct an arbitrary discrete rv defined
      on a finite set of values ``xk`` with ``Prob{X=xk} = pk`` by using the
      ``values`` keyword argument to the `rv_discrete` constructor.
  
      Examples
      --------
  
      Custom made discrete distribution:
  
      >>> from scipy import stats
      >>> xk = np.arange(7)
      >>> pk = (0.1, 0.2, 0.3, 0.1, 0.1, 0.0, 0.2)
      >>> custm = stats.rv_discrete(name='custm', values=(xk, pk))
      >>>
      >>> import matplotlib.pyplot as plt
      >>> fig, ax = plt.subplots(1, 1)
      >>> ax.plot(xk, custm.pmf(xk), 'ro', ms=12, mec='r')
      >>> ax.vlines(xk, 0, custm.pmf(xk), colors='r', lw=4)
      >>> plt.show()
  
      Random number generation:
  
      >>> R = custm.rvs(size=100)
  
      """
  
      def __init__(self, a=0, b=inf, name=None, badvalue=None,
                   moment_tol=1e-8, values=None, inc=1, longname=None,
                   shapes=None, extradoc=None, seed=None):
  
          super(rv_discrete, self).__init__(seed)
  
          # cf generic freeze
          self._ctor_param = dict(
              a=a, b=b, name=name, badvalue=badvalue,
              moment_tol=moment_tol, values=values, inc=inc,
              longname=longname, shapes=shapes, extradoc=extradoc, seed=seed)
  
          if badvalue is None:
              badvalue = nan
          if name is None:
              name = 'Distribution'
          self.badvalue = badvalue
          self.a = a
          self.b = b
          self.name = name
          self.moment_tol = moment_tol
          self.inc = inc
          self._cdfvec = vectorize(self._cdf_single, otypes='d')
          self.return_integers = 1
          self.vecentropy = vectorize(self._entropy)
          self.shapes = shapes
          self.extradoc = extradoc
  
          if values is not None:
              self.xk, self.pk = values
              self.return_integers = 0
              indx = argsort(ravel(self.xk))
              self.xk = take(ravel(self.xk), indx, 0)
              self.pk = take(ravel(self.pk), indx, 0)
              self.a = self.xk[0]
              self.b = self.xk[-1]
              self.P = dict(zip(self.xk, self.pk))
              self.qvals = np.cumsum(self.pk, axis=0)
              self.F = dict(zip(self.xk, self.qvals))
              decreasing_keys = sorted(self.F.keys(), reverse=True)
              self.Finv = dict((self.F[k], k) for k in decreasing_keys)
              self._ppf = instancemethod(vectorize(_drv_ppf, otypes='d'),
                                         self, rv_discrete)
              self._pmf = instancemethod(vectorize(_drv_pmf, otypes='d'),
                                         self, rv_discrete)
              self._cdf = instancemethod(vectorize(_drv_cdf, otypes='d'),
                                         self, rv_discrete)
              self._nonzero = instancemethod(_drv_nonzero, self, rv_discrete)
              self.generic_moment = instancemethod(_drv_moment,
                                                   self, rv_discrete)
              self.moment_gen = instancemethod(_drv_moment_gen,
                                               self, rv_discrete)
  
              self.shapes = ' '   # bypass inspection
              self._construct_argparser(meths_to_inspect=[self._pmf],
                                        locscale_in='loc=0',
                                        # scale=1 for discrete RVs
                                        locscale_out='loc, 1')
          else:
              self._construct_argparser(meths_to_inspect=[self._pmf, self._cdf],
                                        locscale_in='loc=0',
                                        # scale=1 for discrete RVs
                                        locscale_out='loc, 1')
  
              # nin correction needs to be after we know numargs
              # correct nin for generic moment vectorization
              _vec_generic_moment = vectorize(_drv2_moment, otypes='d')
              _vec_generic_moment.nin = self.numargs + 2
              self.generic_moment = instancemethod(_vec_generic_moment,
                                                   self, rv_discrete)
  
              # correct nin for ppf vectorization
              _vppf = vectorize(_drv2_ppfsingle, otypes='d')
              _vppf.nin = self.numargs + 2  # +1 is for self
              self._ppfvec = instancemethod(_vppf,
                                            self, rv_discrete)
  
          # now that self.numargs is defined, we can adjust nin
          self._cdfvec.nin = self.numargs + 1
  
          # generate docstring for subclass instances
          if longname is None:
              if name[0] in ['aeiouAEIOU']:
                  hstr = "An "
              else:
                  hstr = "A "
              longname = hstr + name
  
          if sys.flags.optimize < 2:
              # Skip adding docstrings if interpreter is run with -OO
              if self.__doc__ is None:
                  self._construct_default_doc(longname=longname,
                                              extradoc=extradoc,
                                              docdict=docdict_discrete,
                                              discrete='discrete')
              else:
                  dct = dict(distdiscrete)
                  self._construct_doc(docdict_discrete, dct.get(self.name))
  
              #discrete RV do not have the scale parameter, remove it
              self.__doc__ = self.__doc__.replace(
                  '\n    scale : array_like, '
                  'optional\n        scale parameter (default=1)', '')
  
      def _updated_ctor_param(self):
          """ Return the current version of _ctor_param, possibly updated by user.
  
              Used by freezing and pickling.
              Keep this in sync with the signature of __init__.
          """
          dct = self._ctor_param.copy()
          dct['a'] = self.a
          dct['b'] = self.b
          dct['badvalue'] = self.badvalue
          dct['moment_tol'] = self.moment_tol
          dct['inc'] = self.inc
          dct['name'] = self.name
          dct['shapes'] = self.shapes
          dct['extradoc'] = self.extradoc
          return dct
  
      def _nonzero(self, k, *args):
          return floor(k) == k
  
      def _pmf(self, k, *args):
          return self._cdf(k, *args) - self._cdf(k-1, *args)
  
      def _logpmf(self, k, *args):
          return log(self._pmf(k, *args))
  
      def _cdf_single(self, k, *args):
          m = arange(int(self.a), k+1)
          return np.sum(self._pmf(m, *args), axis=0)
  
      def _cdf(self, x, *args):
          k = floor(x)
          return self._cdfvec(k, *args)
  
      # generic _logcdf, _sf, _logsf, _ppf, _isf, _rvs defined in rv_generic
  
      def rvs(self, *args, **kwargs):
          """
          Random variates of given type.
  
          Parameters
          ----------
          arg1, arg2, arg3,... : array_like
              The shape parameter(s) for the distribution (see docstring of the
              instance object for more information).
          loc : array_like, optional
              Location parameter (default=0).
          size : int or tuple of ints, optional
              Defining number of random variates (Default is 1).  Note that `size`
              has to be given as keyword, not as positional argument.
          random_state : None or int or ``np.random.RandomState`` instance, optional
              If int or RandomState, use it for drawing the random variates.
              If None, rely on ``self.random_state``.
              Default is None.
  
          Returns
          -------
          rvs : ndarray or scalar
              Random variates of given `size`.
  
          """
          kwargs['discrete'] = True
          return super(rv_discrete, self).rvs(*args, **kwargs)
  
      def pmf(self, k, *args, **kwds):
          """
          Probability mass function at k of the given RV.
  
          Parameters
          ----------
          k : array_like
              Quantiles.
          arg1, arg2, arg3,... : array_like
              The shape parameter(s) for the distribution (see docstring of the
              instance object for more information)
          loc : array_like, optional
              Location parameter (default=0).
  
          Returns
          -------
          pmf : array_like
              Probability mass function evaluated at k
  
          """
          args, loc, _ = self._parse_args(*args, **kwds)
          k, loc = map(asarray, (k, loc))
          args = tuple(map(asarray, args))
          k = asarray((k-loc))
          cond0 = self._argcheck(*args)
          cond1 = (k >= self.a) & (k <= self.b) & self._nonzero(k, *args)
          cond = cond0 & cond1
          output = zeros(shape(cond), 'd')
          place(output, (1-cond0) + np.isnan(k), self.badvalue)
          if np.any(cond):
              goodargs = argsreduce(cond, *((k,)+args))
              place(output, cond, np.clip(self._pmf(*goodargs), 0, 1))
          if output.ndim == 0:
              return output[()]
          return output
  
      def logpmf(self, k, *args, **kwds):
          """
          Log of the probability mass function at k of the given RV.
  
          Parameters
          ----------
          k : array_like
              Quantiles.
          arg1, arg2, arg3,... : array_like
              The shape parameter(s) for the distribution (see docstring of the
              instance object for more information).
          loc : array_like, optional
              Location parameter. Default is 0.
  
          Returns
          -------
          logpmf : array_like
              Log of the probability mass function evaluated at k.
  
          """
          args, loc, _ = self._parse_args(*args, **kwds)
          k, loc = map(asarray, (k, loc))
          args = tuple(map(asarray, args))
          k = asarray((k-loc))
          cond0 = self._argcheck(*args)
          cond1 = (k >= self.a) & (k <= self.b) & self._nonzero(k, *args)
          cond = cond0 & cond1
          output = empty(shape(cond), 'd')
          output.fill(NINF)
          place(output, (1-cond0) + np.isnan(k), self.badvalue)
          if np.any(cond):
              goodargs = argsreduce(cond, *((k,)+args))
              place(output, cond, self._logpmf(*goodargs))
          if output.ndim == 0:
              return output[()]
          return output
  
      def cdf(self, k, *args, **kwds):
          """
          Cumulative distribution function of the given RV.
  
          Parameters
          ----------
          k : array_like, int
              Quantiles.
          arg1, arg2, arg3,... : array_like
              The shape parameter(s) for the distribution (see docstring of the
              instance object for more information).
          loc : array_like, optional
              Location parameter (default=0).
  
          Returns
          -------
          cdf : ndarray
              Cumulative distribution function evaluated at `k`.
  
          """
          args, loc, _ = self._parse_args(*args, **kwds)
          k, loc = map(asarray, (k, loc))
          args = tuple(map(asarray, args))
          k = asarray((k-loc))
          cond0 = self._argcheck(*args)
          cond1 = (k >= self.a) & (k < self.b)
          cond2 = (k >= self.b)
          cond = cond0 & cond1
          output = zeros(shape(cond), 'd')
          place(output, (1-cond0) + np.isnan(k), self.badvalue)
          place(output, cond2*(cond0 == cond0), 1.0)
  
          if np.any(cond):
              goodargs = argsreduce(cond, *((k,)+args))
              place(output, cond, np.clip(self._cdf(*goodargs), 0, 1))
          if output.ndim == 0:
              return output[()]
          return output
  
      def logcdf(self, k, *args, **kwds):
          """
          Log of the cumulative distribution function at k of the given RV.
  
          Parameters
          ----------
          k : array_like, int
              Quantiles.
          arg1, arg2, arg3,... : array_like
              The shape parameter(s) for the distribution (see docstring of the
              instance object for more information).
          loc : array_like, optional
              Location parameter (default=0).
  
          Returns
          -------
          logcdf : array_like
              Log of the cumulative distribution function evaluated at k.
  
          """
          args, loc, _ = self._parse_args(*args, **kwds)
          k, loc = map(asarray, (k, loc))
          args = tuple(map(asarray, args))
          k = asarray((k-loc))
          cond0 = self._argcheck(*args)
          cond1 = (k >= self.a) & (k < self.b)
          cond2 = (k >= self.b)
          cond = cond0 & cond1
          output = empty(shape(cond), 'd')
          output.fill(NINF)
          place(output, (1-cond0) + np.isnan(k), self.badvalue)
          place(output, cond2*(cond0 == cond0), 0.0)
  
          if np.any(cond):
              goodargs = argsreduce(cond, *((k,)+args))
              place(output, cond, self._logcdf(*goodargs))
          if output.ndim == 0:
              return output[()]
          return output
  
      def sf(self, k, *args, **kwds):
          """
          Survival function (1 - `cdf`) at k of the given RV.
  
          Parameters
          ----------
          k : array_like
              Quantiles.
          arg1, arg2, arg3,... : array_like
              The shape parameter(s) for the distribution (see docstring of the
              instance object for more information).
          loc : array_like, optional
              Location parameter (default=0).
  
          Returns
          -------
          sf : array_like
              Survival function evaluated at k.
  
          """
          args, loc, _ = self._parse_args(*args, **kwds)
          k, loc = map(asarray, (k, loc))
          args = tuple(map(asarray, args))
          k = asarray(k-loc)
          cond0 = self._argcheck(*args)
          cond1 = (k >= self.a) & (k < self.b)
          cond2 = (k < self.a) & cond0
          cond = cond0 & cond1
          output = zeros(shape(cond), 'd')
          place(output, (1-cond0) + np.isnan(k), self.badvalue)
          place(output, cond2, 1.0)
          if np.any(cond):
              goodargs = argsreduce(cond, *((k,)+args))
              place(output, cond, np.clip(self._sf(*goodargs), 0, 1))
          if output.ndim == 0:
              return output[()]
          return output
  
      def logsf(self, k, *args, **kwds):
          """
          Log of the survival function of the given RV.
  
          Returns the log of the "survival function," defined as 1 - `cdf`,
          evaluated at `k`.
  
          Parameters
          ----------
          k : array_like
              Quantiles.
          arg1, arg2, arg3,... : array_like
              The shape parameter(s) for the distribution (see docstring of the
              instance object for more information).
          loc : array_like, optional
              Location parameter (default=0).
  
          Returns
          -------
          logsf : ndarray
              Log of the survival function evaluated at `k`.
  
          """
          args, loc, _ = self._parse_args(*args, **kwds)
          k, loc = map(asarray, (k, loc))
          args = tuple(map(asarray, args))
          k = asarray(k-loc)
          cond0 = self._argcheck(*args)
          cond1 = (k >= self.a) & (k < self.b)
          cond2 = (k < self.a) & cond0
          cond = cond0 & cond1
          output = empty(shape(cond), 'd')
          output.fill(NINF)
          place(output, (1-cond0) + np.isnan(k), self.badvalue)
          place(output, cond2, 0.0)
          if np.any(cond):
              goodargs = argsreduce(cond, *((k,)+args))
              place(output, cond, self._logsf(*goodargs))
          if output.ndim == 0:
              return output[()]
          return output
  
      def ppf(self, q, *args, **kwds):
          """
          Percent point function (inverse of `cdf`) at q of the given RV.
  
          Parameters
          ----------
          q : array_like
              Lower tail probability.
          arg1, arg2, arg3,... : array_like
              The shape parameter(s) for the distribution (see docstring of the
              instance object for more information).
          loc : array_like, optional
              Location parameter (default=0).
  
          Returns
          -------
          k : array_like
              Quantile corresponding to the lower tail probability, q.
  
          """
          args, loc, _ = self._parse_args(*args, **kwds)
          q, loc = map(asarray, (q, loc))
          args = tuple(map(asarray, args))
          cond0 = self._argcheck(*args) & (loc == loc)
          cond1 = (q > 0) & (q < 1)
          cond2 = (q == 1) & cond0
          cond = cond0 & cond1
          output = valarray(shape(cond), value=self.badvalue, typecode='d')
          # output type 'd' to handle nin and inf
          place(output, (q == 0)*(cond == cond), self.a-1)
          place(output, cond2, self.b)
          if np.any(cond):
              goodargs = argsreduce(cond, *((q,)+args+(loc,)))
              loc, goodargs = goodargs[-1], goodargs[:-1]
              place(output, cond, self._ppf(*goodargs) + loc)
  
          if output.ndim == 0:
              return output[()]
          return output
  
      def isf(self, q, *args, **kwds):
          """
          Inverse survival function (inverse of `sf`) at q of the given RV.
  
          Parameters
          ----------
          q : array_like
              Upper tail probability.
          arg1, arg2, arg3,... : array_like
              The shape parameter(s) for the distribution (see docstring of the
              instance object for more information).
          loc : array_like, optional
              Location parameter (default=0).
  
          Returns
          -------
          k : ndarray or scalar
              Quantile corresponding to the upper tail probability, q.
  
          """
          args, loc, _ = self._parse_args(*args, **kwds)
          q, loc = map(asarray, (q, loc))
          args = tuple(map(asarray, args))
          cond0 = self._argcheck(*args) & (loc == loc)
          cond1 = (q > 0) & (q < 1)
          cond2 = (q == 1) & cond0
          cond = cond0 & cond1
  
          # same problem as with ppf; copied from ppf and changed
          output = valarray(shape(cond), value=self.badvalue, typecode='d')
          # output type 'd' to handle nin and inf
          place(output, (q == 0)*(cond == cond), self.b)
          place(output, cond2, self.a-1)
  
          # call place only if at least 1 valid argument
          if np.any(cond):
              goodargs = argsreduce(cond, *((q,)+args+(loc,)))
              loc, goodargs = goodargs[-1], goodargs[:-1]
              # PB same as ticket 766
              place(output, cond, self._isf(*goodargs) + loc)
  
          if output.ndim == 0:
              return output[()]
          return output
  
      def _entropy(self, *args):
          if hasattr(self, 'pk'):
              return entropy(self.pk)
          else:
              return _expect(lambda x: entr(self.pmf(x, *args)),
                             self.a, self.b, self.ppf(0.5, *args), self.inc)
  
      def expect(self, func=None, args=(), loc=0, lb=None, ub=None,
                 conditional=False, maxcount=1000, tolerance=1e-10, chunksize=32):
          """
          Calculate expected value of a function with respect to the distribution
          for discrete distribution.
  
          Parameters
          ----------
          func : callable, optional
              Function for which the expectation value is calculated.
              Takes only one argument.
              The default is the identity mapping f(k) = k.
          args : tuple, optional
              Shape parameters of the distribution.
          loc : float, optional
              Location parameter.
              Default is 0.
          lb, ub : int, optional
              Lower and upper bound for the summation, default is set to the
              support of the distribution, inclusive (``ul <= k <= ub``).
          conditional : bool, optional
              If true then the expectation is corrected by the conditional
              probability of the summation interval. The return value is the
              expectation of the function, `func`, conditional on being in
              the given interval (k such that ``ul <= k <= ub``).
              Default is False.
          maxcount : int, optional
              Maximal number of terms to evaluate (to avoid an endless loop for
              an infinite sum). Default is 1000.
          tolerance : float, optional
              Absolute tolerance for the summation. Default is 1e-10.
          chunksize : int, optional
              Iterate over the support of a distributions in chunks of this size.
              Default is 32.
  
          Returns
          -------
          expect : float
              Expected value.
  
          Notes
          -----
          For heavy-tailed distributions, the expected value may or may not exist,
          depending on the function, `func`. If it does exist, but the sum converges
          slowly, the accuracy of the result may be rather low. For instance, for
          ``zipf(4)``, accuracy for mean, variance in example is only 1e-5.
          increasing `maxcount` and/or `chunksize` may improve the result, but may also
          make zipf very slow.
  
          The function is not vectorized.
  
          """
          if func is None:
              def fun(x):
                  # loc and args from outer scope
                  return (x+loc)*self._pmf(x, *args)
          else:
              def fun(x):
                  # loc and args from outer scope
                  return func(x+loc)*self._pmf(x, *args)
          # used pmf because _pmf does not check support in randint and there
          # might be problems(?) with correct self.a, self.b at this stage maybe
          # not anymore, seems to work now with _pmf
  
          self._argcheck(*args)  # (re)generate scalar self.a and self.b
          if lb is None:
              lb = self.a
          else:
              lb = lb - loc   # convert bound for standardized distribution
          if ub is None:
              ub = self.b
          else:
              ub = ub - loc   # convert bound for standardized distribution
          if conditional:
              invfac = self.sf(lb-1, *args) - self.sf(ub, *args)
          else:
              invfac = 1.0
  
          # iterate over the support, starting from the median
          x0 = self.ppf(0.5, *args)
          res = _expect(fun, lb, ub, x0, self.inc, maxcount, tolerance, chunksize)
          return res / invfac
  
  
  def _expect(fun, lb, ub, x0, inc, maxcount=1000, tolerance=1e-10,
              chunksize=32):
      """Helper for computing the expectation value of `fun`."""
  
      # short-circuit if the support size is small enough
      if (ub - lb) <= chunksize:
          supp = np.arange(lb, ub+1, inc)
          vals = fun(supp)
          return np.sum(vals)
  
      # otherwise, iterate starting from x0
      if x0 < lb:
          x0 = lb
      if x0 > ub:
          x0 = ub
  
      count, tot = 0, 0.
      # iterate over [x0, ub] inclusive
      for x in _iter_chunked(x0, ub+1, chunksize=chunksize, inc=inc):
          count += x.size
          delta = np.sum(fun(x))
          tot += delta
          if abs(delta) < tolerance * x.size:
              break
          if count > maxcount:
              warnings.warn('expect(): sum did not converge', RuntimeWarning)
              return tot
  
      # iterate over [lb, x0)
      for x in _iter_chunked(x0-1, lb-1, chunksize=chunksize, inc=-inc):
          count += x.size
          delta = np.sum(fun(x))
          tot += delta
          if abs(delta) < tolerance * x.size:
              break
          if count > maxcount:
              warnings.warn('expect(): sum did not converge', RuntimeWarning)
              break
  
      return tot
  
  
  def _iter_chunked(x0, x1, chunksize=4, inc=1):
      """Iterate from x0 to x1 in chunks of chunksize and steps inc.
  
      x0 must be finite, x1 need not be. In the latter case, the iterator is infinite.
      Handles both x0 < x1 and x0 > x1. In the latter case, iterates downwards
      (make sure to set inc < 0.)
  
      >>> [x for x in _iter_chunked(2, 5, inc=2)]
      [array([2, 4])]
      >>> [x for x in _iter_chunked(2, 11, inc=2)]
      [array([2, 4, 6, 8]), array([10])]
      >>> [x for x in _iter_chunked(2, -5, inc=-2)]
      [array([ 2,  0, -2, -4])]
      >>> [x for x in _iter_chunked(2, -9, inc=-2)]
      [array([ 2,  0, -2, -4]), array([-6, -8])]
  
      """
      if inc == 0:
          raise ValueError('Cannot increment by zero.')
      if chunksize <= 0:
          raise ValueError('Chunk size must be positive; got %s.' % chunksize)
  
      s = 1 if inc > 0 else -1
      stepsize = abs(chunksize * inc)
  
      x = x0
      while (x - x1) * inc < 0:
          delta = min(stepsize, abs(x - x1))
          step = delta * s
          supp = np.arange(x, x + step, inc)
          x += step
          yield supp
  
  
  def get_distribution_names(namespace_pairs, rv_base_class):
      """
      Collect names of statistical distributions and their generators.
  
      Parameters
      ----------
      namespace_pairs : sequence
          A snapshot of (name, value) pairs in the namespace of a module.
      rv_base_class : class
          The base class of random variable generator classes in a module.
  
      Returns
      -------
      distn_names : list of strings
          Names of the statistical distributions.
      distn_gen_names : list of strings
          Names of the generators of the statistical distributions.
          Note that these are not simply the names of the statistical
          distributions, with a _gen suffix added.
  
      """
      distn_names = []
      distn_gen_names = []
      for name, value in namespace_pairs:
          if name.startswith('_'):
              continue
          if name.endswith('_gen') and issubclass(value, rv_base_class):
              distn_gen_names.append(name)
          if isinstance(value, rv_base_class):
              distn_names.append(name)
      return distn_names, distn_gen_names
  
apis: numpy.sum scipy.misc.doccer.docformat numpy.ones numpy.vectorize numpy.any numpy.asarray numpy.log scipy.misc.derivative numpy.isscalar numpy.isfinite numpy.extract scipy._lib.six.exec_ numpy.reshape scipy._lib._util.getargspec_no_self numpy.seterr numpy.isnan scipy.special.entr scipy.special.xlogy scipy._lib._util.check_random_state numpy.argmax numpy.arange numpy.power numpy.all numpy.min numpy.max scipy.special.comb numpy.array scipy.integrate.quad numpy.cumsum numpy.isinf numpy.floor numpy.exp numpy.atleast_1d numpy.ravel scipy.optimize.brentq numpy.place numpy.shape numpy.product scipy.special.chndtr numpy.isneginf numpy.sqrt numpy.find_common_type scipy.special.kl_div scipy.special.ive