  We extend models of our Galaxy based on distribution functions (DFs) that are
analytic functions of the action integrals to extended distribution functions
(EDFs), which have an analytic dependence on metallicity as well. We use a
simple, but physically-motivated, functional forms for the metallicity of the
interstellar medium as a function of radius and time and for the star-formation
rate, and a model for the diffusion of stars through phase space to suggest the
required functional form of an EDF. We introduce a simple prescription for
radial migration that preserves the overall profile of the disc while allowing
individual stars to migrate throughout the disc. Our models explicitly consider
the thin and thick discs as two distinct components separated in age.
  We show how an EDF can be used to incorporate realistic selection functions
in models, and to construct mock catalogues of observed samples. We show that
the selection function of the Geneva-Copenhagen Survey (GCS) biases in favour
of young stars, which have atypically small random velocities. With the
selection function taken into account our models produce good fits of the GCS
data in chemo-dynamical space and the Gilmore and Reid (1983) density data.
  From our EDF, we predict the structure of the SEGUE G-dwarf sample. The
kinematics are successfully predicted. The predicted metallicity distribution
has too few stars with [Fe/H]$\simeq-0.5$ dex and too many metal-rich stars. A
significant problem may be the lack of any chemical-kinematic correlations in
our thick disc. We argue that EDFs will prove essential tools for the analysis
of both observational data and sophisticated models of Galaxy formation and
evolution.
