.

        other: a number

        returns: new Pmf
        """
        pmf = Pmf()
        for v1, p1 in self.Items():
            pmf.Set(v1 + other, p1)
        return pmf

    def __sub__(self, other):
        """Computes the Pmf of the diff of values drawn from self and other.

        other: another Pmf

        returns: new Pmf
        """
        try:
            return self.SubPmf(other)
        except AttributeError:
            return self.AddConstant(-other)

    def SubPmf(self, other):
        """Computes the Pmf of the diff of values drawn from self and other.

        other: another Pmf

        returns: new Pmf
        """
        pmf = Pmf()
        for v1, p1 in self.Items():
            for v2, p2 in other.Items():
                pmf.Incr(v1 - v2, p1 * p2)
        return pmf

    def __mul__(self, other):
        """Computes the Pmf of the product of values drawn from self and other.

        other: another Pmf

        returns: new Pmf
        """
        try:
            return self.MulPmf(other)
        except AttributeError:
            return self.MulConstant(other)

    def MulPmf(self, other):
        """Computes the Pmf of the diff of values drawn from self and other.

        other: another Pmf

        returns: new Pmf
        """
        pmf = Pmf()
        for v1, p1 in self.Items():
            for v2, p2 in other.Items():
                pmf.Incr(v1 * v2, p1 * p2)
        return pmf

    def MulConstant(self, other):
        """Computes the Pmf of the product of a constant and values from self.

        other: a number

        returns: new Pmf
        """
        pmf = Pmf()
        for v1, p1 in self.Items():
            pmf.Set(v1 * other, p1)
        return pmf

    def __div__(self, other):
        """Computes the Pmf of the ratio of values drawn from self and other.

        other: another Pmf

        returns: new Pmf
        """
        try:
            return self.DivPmf(other)
        except AttributeError:
            return self.MulConstant(1/other)

    __truediv__ = __div__

    def DivPmf(self, other):
        """Computes the Pmf of the ratio of values drawn from self and other.

        other: another Pmf

        returns: new Pmf
        """
        pmf = Pmf()
        for v1, p1 in self.Items():
            for v2, p2 in other.Items():
                pmf.Incr(v1 / v2, p1 * p2)
        return pmf

    def Max(self, k):
        """Computes the CDF of the maximum of k selections from this dist.

        k: int

        returns: new Cdf
        """
        cdf = self.MakeCdf()
        return cdf.Max(k)


class Joint(Pmf):
    """Represents a joint distribution.

    The values are sequences (usually tuples)
    """

    def Marginal(self, i, label=None):
        """Gets the marginal distribution of the indicated variable.

        i: index of the variable we want

        Returns: Pmf
        """
        pmf = Pmf(label=label)
        for vs, prob in self.Items():
            pmf.Incr(vs[i], prob)
        return pmf

    def Conditional(self, i, j, val, label=None):
        """Gets the conditional distribution of the indicated variable.

        Distribution of vs[i], conditioned on vs[j] = val.

        i: index of the variable we want
        j: which variable is conditioned on
        val: the value the jth variable has to have

        Returns: Pmf
        """
        pmf = Pmf(label=label)
        for vs, prob in self.Items():
            if vs[j] != val:
                continue
            pmf.Incr(vs[i], prob)

        pmf.Normalize()
        return pmf

    def MaxLikeInterval(self, percentage=90):
        """Returns the maximum-likelihood credible interval.

        If percentage=90, computes a 90% CI containing the values
        with the highest likelihoods.

        percentage: float between 0 and 100

        Returns: list of values from the suite
        """
        interval = []
        total = 0

        t = [(prob, val) for val, prob in self.Items()]
        t.sort(reverse=True)

        for prob, val in t:
            interval.append(val)
            total += prob
            if total >= percentage / 100.0:
                break

        return interval


def MakeJoint(pmf1, pmf2):
    """Joint distribution of values from pmf1 and pmf2.

    Assumes that the PMFs represent independent random variables.

    Args:
        pmf1: Pmf object
        pmf2: Pmf object

    Returns:
        Joint pmf of value pairs
    """
    joint = Joint()
    for v1, p1 in pmf1.Items():
        for v2, p2 in pmf2.Items():
            joint.Set((v1, v2), p1 * p2)
    return joint


def MakeHistFromList(t, label=None):
    """Makes a histogram from an unsorted sequence of values.

    Args:
        t: sequence of numbers
        label: string label for this histogram

    Returns:
        Hist object
    """
    return Hist(t, label=label)


def MakeHistFromDict(d, label=None):
    """Makes a histogram from a map from values to frequencies.

    Args:
        d: dictionary that maps values to frequencies
        label: string label for this histogram

    Returns:
        Hist object
    """
    return Hist(d, label)


def MakePmfFromList(t, label=None):
    """Makes a PMF from an unsorted sequence of values.

    Args:
        t: sequence of numbers
        label: string label for this PMF

    Returns:
        Pmf object
    """
    return Pmf(t, label=label)


def MakePmfFromDict(d, label=None):
    """Makes a PMF from a map from values to probabilities.

    Args:
        d: dictionary that maps values to probabilities
        label: string label for this PMF

    Returns:
        Pmf object
    """
    return Pmf(d, label=label)


def MakePmfFromItems(t, label=None):
    """Makes a PMF from a sequence of value-probability pairs

    Args:
        t: sequence of value-probability pairs
        label: string label for this PMF

    Returns:
        Pmf object
    """
    return Pmf(dict(t), label=label)


def MakePmfFromHist(hist, label=None):
    """Makes a normalized PMF from a Hist object.

    Args:
        hist: Hist object
        label: string label

    Returns:
        Pmf object
    """
    if label is None:
        label = hist.label

    return Pmf(hist, label=label)


def MakeMixture(metapmf, label='mix'):
    """Make a mixture distribution.

    Args:
      metapmf: Pmf that maps from Pmfs to probs.
      label: string label for the new Pmf.

    Returns: Pmf object.
    """
    mix = Pmf(label=label)
    for pmf, p1 in metapmf.Items():
        for x, p2 in pmf.Items():
            mix.Incr(x, p1 * p2)
    return mix


def MakeUniformPmf(low, high, n):
    """Make a uniform Pmf.

    low: lowest value (inclusive)
    high: highest value (inclusize)
    n: number of values
    """
    pmf = Pmf()
    for x in np.linspace(low, high, n):
        pmf.Set(x, 1)
    pmf.Normalize()
    return pmf


class Cdf(object):
    """Represents a cumulative distribution function.

    Attributes:
        xs: sequence of values
        ps: sequence of probabilities
        label: string used as a graph label.
    """
    def __init__(self, obj=None, ps=None, label=None):
        """Initializes.
        
        If ps is provided, obj must be the corresponding list of values.

        obj: Hist, Pmf, Cdf, Pdf, dict, pandas Series, list of pairs
        ps: list of cumulative probabilities
        label: string label
        """
        self.label = label if label is not None else '_nolegend_'

        if isinstance(obj, (_DictWrapper, Cdf, Pdf)):
            if not label:
                self.label = label if label is not None else obj.label

        if obj is None:
            # caller does not provide obj, make an empty Cdf
            self.xs = np.asarray([])
            self.ps = np.asarray([])
            if ps is not None:
                logging.warning("Cdf: can't pass ps without also passing xs.")
            return
        else:
            # if the caller provides xs and ps, just store them          
            if ps is not None:
                if isinstance(ps, str):
                    logging.warning("Cdf: ps can't be a string")

                self.xs = np.asarray(obj)
                self.ps = np.asarray(ps)
                return

        # caller has provided just obj, not ps
        if isinstance(obj, Cdf):
            self.xs = copy.copy(obj.xs)
            self.ps = copy.copy(obj.ps)
            return

        if isinstance(obj, _DictWrapper):
            dw = obj
        else:
            dw = Hist(obj)

        if len(dw) == 0:
            self.xs = np.asarray([])
            self.ps = np.asarray([])
            return

        xs, freqs = zip(*sorted(dw.Items()))
        self.xs = np.asarray(xs)
        self.ps = np.cumsum(freqs, dtype=np.float)
        self.ps /= self.ps[-1]

    def __str__(self):
        return 'Cdf(%s, %s)' % (str(self.xs), str(self.ps))

    __repr__ = __str__

    def __len__(self):
        return len(self.xs)

    def __getitem__(self, x):
        return self.Prob(x)

    def __setitem__(self):
        raise UnimplementedMethodException()

    def __delitem__(self):
        raise UnimplementedMethodException()

    def __eq__(self, other):
        return np.all(self.xs == other.xs) and np.all(self.ps == other.ps)

    def Copy(self, label=None):
        """Returns a copy of this Cdf.

        label: string label for the new Cdf
        """
        if label is None:
            label = self.label
        return Cdf(list(self.xs), list(self.ps), label=label)

    def MakePmf(self, label=None):
        """Makes a Pmf."""
        if label is None:
            label = self.label
        return Pmf(self, label=label)

    def Values(self):
        """Returns a sorted list of values.
        """
        return self.xs

    def Items(self):
        """Returns a sorted sequence o