  In this work, we study the short time dynamics of a molecular junction
described by Anderson-Holstein model using full-counting statistics after
projective measurement. The coupling between the central quantum dot (QD) and
two leads was turned on at remote past and the system is evolved to steady
state at time $t=0$, when we perform the projective measurement in one of the
lead. Generating function for the charge transfer is expressed as a Fredholm
determinant in terms of Keldysh nonequilibrium Green's function in the time
domain. It is found that the current is not constant at short times indicating
that the measurement does perturb the system. We numerically compare the
current behaviors after the projective measurement with those in the transient
regime where the subsystems are connected at $t=0$. The universal scaling for
high-order cumulants is observed for the case with zero QD occupation due to
the unidirectional transport at short times. The influences of electron-phonon
interaction on short time dynamics of electric current, shot noise and
differential conductance are analyzed.
