  In this paper, we show two kinds of entangled many body systems with special
statistic properties. Firstly, an entangled fermions system with a pairwise
entanglement between every two particles in the lowest energy energy level
obeys the fractional statistics. As a check, for particle number N=2, N=3 and
N=4, considering that any two fermions in the lowest Landau level are entangled
in a proper way, the Laughlin wave function can be derived. The results reveals
the explicit entanglement pattern of the Laughlin states. Secondly, we noticed
that both Bose-Einstein statistics and Fermi-Dirac distributions are derived
from computing the partial function of a free quantum many body system in a
certain ensemble without considering entanglement. We extend the computation of
the partial function to an entangled quantum many body system without
interaction, in this system we assume that every particle in energy level
$\epsilon_i$ is entangled with a particle in the energy level $\epsilon_{i+1}$
($i=1,3,5,...$) and also every particle in energy level $\epsilon_i+1$ is
entangled with a particle in the energy level $\epsilon_{i}$ ($i=1,3,5,...$),
which indicates that the two energy level have the same number of particles. In
the entangled system, we find that the partial function will be changed. As a
results, both the Bose-Einstein Statics and the Fermi-Dirac distributions will
be modified at finite temperature.
