  A potential experimental probe, forward-backward elliptic anisotropy
correlation ($C_{FB} $), has been proposed by Liao and Koch to distinguish the
jet and true elliptic flow contribution to the measured elliptic flow ($v_2$)
in relativistic heavy-ion collisions. Jet and flow fluctuation contribution to
elliptic flow is investigated within the framework of a multi-phase transport
model using the $C_{FB} $ probe. We found that the $C_{FB} $ correlation is
remarkably different and is about two times of that proposed by Liao and Koch.
It originates from the correlation between fluctuation of forward and backward
elliptic flow at low transverse momentum, which is mainly due to the initial
correlation between fluctuation of forward and backward eccentricity. This
results in an amendment of the $C_{FB}$ by a term related to the correlation
between fluctuation of forward and backward elliptic flow. Our results suggest
that a suitable rapidity gap for $C_{FB} $ correlation studies should be around
$\pm$ 3.5.
