  We extend the study of lowest moments, $<x>$ and $<x^2>$, of the parton
distribution function of the nucleon to include those of the sea quarks; this
entails a disconnected insertion calculation in lattice QCD. This is carried
out on a $16^3 \times 24$ quenched lattice with Wilson fermion. The quark loops
are calculated with $Z_2$ noise vectors and unbiased subtractions, and multiple
nucleon sources are employed to reduce the statistical errors. We obtain
5$\sigma$ signals for $<x>$ for the $u,d,$ and $s$ quarks, but $<x^2>$ is
consistent with zero within errors. We provide results for both the connected
and disconnected insertions. The perturbatively renormalized $<x>$ for the
strange quark at $\mu = 2$ GeV is $<x>_{s+\bar{s}} = 0.027 \pm 0.006$ which is
consistent with the experimental result. The ratio of $<x>$ for $s$ vs. $u/d$
in the disconnected insertion with quark loops is calculated to be $0.88 \pm
0.07$. This is about twice as large as the phenomenologically fitted
$\displaystyle\frac{< x>_{s+\bar{s}}}{< x>_{\bar{u}}+< x>_{\bar{d}}}$ from
experiments where $\bar{u}$ and $\bar{d}$ include both the connected and
disconnected insertion parts. We discuss the source and implication of this
difference.
