code: def funsor_to_cat_and_mvn(funsor_, ndims, event_inputs):
      """
      Converts a labeled gaussian mixture model to a pair of distributions.
  
      :param funsor.joint.Joint funsor_: A Gaussian mixture funsor.
      :param int ndims: The number of batch dimensions in the result.
      :return: A pair ``(cat, mvn)``, where ``cat`` is a
          :class:`~pyro.distributions.Categorical` distribution over mixture
          components and ``mvn`` is a
          :class:`~pyro.distributions.MultivariateNormal` with rightmost batch
          dimension ranging over mixture components.
      """
      assert isinstance(funsor_, Contraction), funsor_
      assert (
          sum(1 for d in funsor_.inputs.values() if d.dtype == "real")
          == 1
      )
      assert event_inputs, "no components name found"
      assert not any(isinstance(v, Delta) for v in funsor_.terms)
      cat, mvn = to_data(
          funsor_, name_to_dim=default_name_to_dim(event_inputs)
      )
      if ndims != len(cat.batch_shape):
          cat = cat.expand(
              (1,) * (ndims - len(cat.batch_shape)) + cat.batch_shape
          )
      if ndims + 1 != len(mvn.batch_shape):
          mvn = mvn.expand(
              (1,) * (ndims + 1 - len(mvn.batch_shape))
              + mvn.batch_shape
          )
      return cat, mvn
  
imports: import math
  from collections import OrderedDict
  import torch
  import funsor.ops
  from funsor.cnf import Contraction
  from funsor.delta import Delta
  from funsor.domains import Array, Bint, Real, Reals
  from funsor.gaussian import Gaussian
  from funsor.interpreter import gensym
  from funsor.tensor import Tensor
  from funsor.terms import Independent, Variable, to_data, to_funsor
label_desc: Use image node_id 8 for calling a global function with example usage: funsor_to_cat_and_mvn(funsor_, ndims, event_inputs) and returns: cat,  mvn