text: """
  Finite Discrete Random Variables Module
  
  See Also
  ========
  diofant.stats.frv_types
  diofant.stats.rv
  diofant.stats.crv
  
  """
  
  import random
  from itertools import product
  
  from ..core import Dict, Eq, Expr, Lambda, Mul, Symbol, Tuple, cacheit, sympify
  from ..functions import Piecewise
  from ..logic import And, Or
  from ..sets import FiniteSet
  from .rv import (ConditionalDomain, NamedArgsMixin, ProductDomain,
                   ProductPSpace, PSpace, RandomDomain, SinglePSpace,
                   random_symbols, rv_subs, sumsets)
  
  
  class FiniteDensity(dict):
      """Finite probabibility density."""
  
      def __call__(self, item):
          item = sympify(item)
          if item in self:
              return self[item]
          else:
              return 0
  
      @property
      def dict(self):
          return dict(self)
  
  
  class FiniteDomain(RandomDomain):
      """
      A domain with discrete finite support
  
      Represented using a FiniteSet.
  
      """
  
      is_Finite = True
  
      @property
      def dict(self):
          return FiniteSet(*[Dict(dict(el)) for el in self.elements])
  
      def as_boolean(self):
          return Or(*[And(*[Eq(sym, val) for sym, val in item]) for item in self])
  
      def __iter__(self):
          raise NotImplementedError
  
  
  class SingleFiniteDomain(FiniteDomain):
      """
      A FiniteDomain over a single symbol/set
  
      Example: The possibilities of a *single* die roll.
  
      """
  
      def __new__(cls, symbol, set):
          if not isinstance(set, FiniteSet):
              set = FiniteSet(*set)
          return Expr.__new__(cls, symbol, set)
  
      @property
      def symbol(self):
          return self.args[0]
  
      @property
      def symbols(self):
          return FiniteSet(self.symbol)
  
      @property
      def set(self):
          return self.args[1]
  
      @property
      def elements(self):
          return FiniteSet(*[frozenset(((self.symbol, elem), )) for elem in self.set])
  
      def __iter__(self):
          return (frozenset(((self.symbol, elem),)) for elem in self.set)
  
      def __contains__(self, other):
          sym, val = tuple(other)[0]
          return sym == self.symbol and val in self.set
  
  
  class ProductFiniteDomain(ProductDomain, FiniteDomain):
      """
      A Finite domain consisting of several other FiniteDomains
  
      Example: The possibilities of the rolls of three independent dice
  
      """
  
      def __iter__(self):
          proditer = product(*self.domains)
          return (sumsets(items) for items in proditer)
  
      @property
      def elements(self):
          return FiniteSet(*self)
  
  
  class ConditionalFiniteDomain(ConditionalDomain, ProductFiniteDomain):
      """
      A FiniteDomain that has been restricted by a condition
  
      Example: The possibilities of a die roll under the condition that the
      roll is even.
  
      """
  
      def __new__(cls, domain, condition):
          if condition is True:
              return domain
          cond = rv_subs(condition)
          # Check that we aren't passed a condition like die1 == z
          # where 'z' is a symbol that we don't know about
          # We will never be able to test this equality through iteration
          if not cond.free_symbols.issubset(domain.free_symbols):
              raise ValueError('Condition "%s" contains foreign symbols \n%s.\n' % (  # noqa: SFS101
                  condition, tuple(cond.free_symbols - domain.free_symbols)) +
                  'Will be unable to iterate using this condition')
  
          return Expr.__new__(cls, domain, cond)
  
      def _test(self, elem):
          val = self.condition.xreplace(dict(elem))
          if val in [True, False]:
              return val
          elif val.is_Equality:
              return val.lhs == val.rhs
          raise ValueError(f'Undeciable if {val!s}')
  
      def __contains__(self, other):
          return other in self.fulldomain and self._test(other)
  
      def __iter__(self):
          return (elem for elem in self.fulldomain if self._test(elem))
  
      @property
      def set(self):
          if self.fulldomain.__class__ is SingleFiniteDomain:
              return FiniteSet(*[elem for elem in self.fulldomain.set
                                 if frozenset(((self.fulldomain.symbol, elem),)) in self])
          else:
              raise NotImplementedError(
                  'Not implemented on multi-dimensional conditional domain')
  
      def as_boolean(self):
          return FiniteDomain.as_boolean(self)
  
  
  class SingleFiniteDistribution(Expr, NamedArgsMixin):
      """Base class for finite distributions."""
  
      def __new__(cls, *args):
          args = list(map(sympify, args))
          return Expr.__new__(cls, *args)
  
      @property  # type: ignore[misc]
      @cacheit
      def dict(self):
          return {k: self.pdf(k) for k in self.set}
  
      @property
      def pdf(self):
          x = Symbol('x')
          return Lambda(x, Piecewise(*(
              [(v, Eq(k, x)) for k, v in self.dict.items()] + [(0, True)])))
  
      @property
      def set(self):
          return list(self.dict)
  
      values = property(lambda self: self.dict.values)
      items = property(lambda self: self.dict.items)
      __iter__ = property(lambda self: self.dict.__iter__)
      __getitem__ = property(lambda self: self.dict.__getitem__)
  
      __call__ = pdf
  
      def __contains__(self, other):
          return other in self.set
  
  
  #####################
  # Probability Space #
  #####################
  
  
  class FinitePSpace(PSpace):
      """
      A Finite Probability Space
  
      Represents the probabilities of a finite number of events.
  
      """
  
      is_Finite = True
  
      @property
      def domain(self):
          return self.args[0]
  
      @property
      def density(self):
          return self.args[0]
  
      def __new__(cls, domain, density):
          density = {sympify(key): sympify(val)
                     for key, val in density.items()}
          public_density = Dict(density)
  
          obj = PSpace.__new__(cls, domain, public_density)
          obj._density = density
          return obj
  
      def prob_of(self, elem):
          return self._density.get(elem, 0)
  
      def where(self, condition):
          return ConditionalFiniteDomain(self.domain, condition)
  
      def compute_density(self, expr):
          expr = expr.xreplace({rs: rs.symbol for rs in self.values})
          d = FiniteDensity()
          for elem in self.domain:
              val = expr.xreplace(dict(elem))
              prob = self.prob_of(elem)
              d[val] = d.get(val, 0) + prob
          return d
  
      @cacheit
      def compute_cdf(self, expr):
          d = self.compute_density(expr)
          cum_prob = 0
          cdf = []
          for key in sorted(d):
              prob = d[key]
              cum_prob += prob
              cdf.append((key, cum_prob))
  
          return dict(cdf)
  
      @cacheit
      def sorted_cdf(self, expr, python_float=False):
          cdf = self.compute_cdf(expr)
          items = list(cdf.items())
          sorted_items = sorted(items, key=lambda val_cumprob: val_cumprob[1])
          if python_float:
              sorted_items = [(v, float(cum_prob))
                              for v, cum_prob in sorted_items]
          return sorted_items
  
      def integrate(self, expr, rvs=None):
          rvs = rvs or self.values
          expr = expr.xreplace({rs: rs.symbol for rs in rvs})
          return sum(expr.xreplace(dict(elem)) * self.prob_of(elem)
                     for elem in self.domain)
  
      def probability(self, condition):
          cond_symbols = frozenset(rs.symbol for rs in random_symbols(condition))
          assert cond_symbols.issubset(self.symbols)
          return sum(self.prob_of(elem) for elem in self.where(condition))
  
      def conditional_space(self, condition):
          domain = self.where(condition)
          prob = self.probability(condition)
          density = {key: val / prob
                     for key, val in self._density.items() if key in domain}
          return FinitePSpace(domain, density)
  
      def sample(self):
          """
          Internal sample method
  
          Returns dictionary mapping RandomSymbol to realization value.
  
          """
          expr = Tuple(*self.values)
          cdf = self.sorted_cdf(expr, python_float=True)
  
          x = random.uniform(0, 1)
          # Find first occurrence with cumulative probability less than x
          # This should be replaced with binary search
          for value, cum_prob in cdf:  # pragma: no branch
              if x < cum_prob:
                  # return dictionary mapping RandomSymbols to values
                  return dict(zip(expr, value))
  
  
  class SingleFinitePSpace(SinglePSpace, FinitePSpace):
      """
      A single finite probability space
  
      Represents the probabilities of a set of random events that can be
      attributed to a single variable/symbol.
  
      This class is implemented by many of the standard FiniteRV types such as
      Die, Bernoulli, Coin, etc....
  
      """
  
      @property
      def domain(self):
          return SingleFiniteDomain(self.symbol, self.distribution.set)
  
      @property  # type: ignore[misc]
      @cacheit
      def _density(self):
          return {frozenset(((self.symbol, val),)): prob
                  for val, prob in self.distribution.dict.items()}
  
  
  class ProductFinitePSpace(ProductPSpace, FinitePSpace):
      """A collection of several independent finite probability spaces."""
  
      @property
      def domain(self):
          return ProductFiniteDomain(*[space.domain for space in self.spaces])
  
      @property  # type: ignore[misc]
      @cacheit
      def _density(self):
          proditer = product(*[iter(space._density.items())
                               for space in self.spaces])
          d = {}
          for items in proditer:
              elems, probs = list(zip(*items))
              elem = sumsets(elems)
              prob = Mul(*probs)
              d[elem] = d.get(elem, 0) + prob
          return Dict(d)
  
      @property  # type: ignore[misc]
      @cacheit
      def density(self):
          return self._density
  