Cross-correlation of the 2XMMi catalogue with Data Release 7 of the
  Sloan Digital Sky Survey
  The Survey Science Centre of the XMM-Newton satellite released the first
incremental version of the 2XMM catalogue in August 2008 . With more than
220,000 X-ray sources, the 2XMMi was at that time the largest catalogue of
X-ray sources ever published and thus constitutes an unprecedented resource for
studying the high-energy properties of various classes of X-ray emitters such
as AGN and stars. The advent of the 7th release of the Sloan Digital Sky Survey
offers the opportunity to cross-match two major surveys and extend the spectral
energy distribution of many 2XMMi sources towards the optical bands. We here
present a cross-matching algorithm based on the classical likelihood ratio
estimator. The method developed has the advantage of providing true
probabilities of identifications without resorting to Monte-Carlo simulations.
Over 30,000 2XMMi sources have SDSS counterparts with individual probabilities
of identification higher than 90%. Using spectroscopic identifications from the
SDSS DR7 catalogue supplemented by extraction from other catalogues, we build
an identified sample from which the way the various classes of X-ray emitters
gather in the multi dimensional parameter space can be analysed. We investigate
two scientific use cases. In the first example we show how these
multi-wavelength data can be used to search for new QSO2s. Although no specific
range of observed properties allows us to identify Compton Thick QSO2s, we show
that the prospects are much better for Compton Thin AGN2 and discuss several
possible multi-parameter selection strategies. In a second example, we confirm
the hardening of the mean X-ray spectrum with increasing X-ray luminosity on a
sample of over 500 X-ray active stars and reveal that on average X-ray active M
stars display bluer $g-r$ colour indexes than less active ones (abridged).
