  By using the determinant Quantum Monte Carlo method, the magnetic and pairing
correlation of the Na$_{x}$CoO$_{2}\cdot$yH$_{2}$O system are studied within
the Hubbard model on a bilayer triangular lattice. The temperature dependence
of spin correlation function and pairing susceptibility with several kinds of
symmetries at different electron fillings and inter layer coupling terms are
investigated. It is found that the system shows an antiferromagnetic
correlation around the half filling, and the $fn$-wave pairing correlation
dominates over other kinds of pairing symmetry in the low doping region. As the
electron filling decreases away from the half filling, both the ferromagnetic
correlation and the $f$-wave paring susceptibility are enhanced and tend to
dominate. It is also shown that both the magnetic susceptibility and paring
susceptibility decrease as the inter layer coupling increases.
