  The shock-acceleration theory predicts a power-law energy spectrum in the
test particle approximation, and there are two ways to calculate a power-law
index, Peacock's approximation and Vietri's formulation. In Peacock's
approximation, it is assumed that particles cross a shock front many times and
energy-gains for each step are fully uncorrelated. On the other hand,
correlation of the distribution of an energy-gain factor for a particle is
considered in Vietri's formulation. We examine how Peacock's approximation
differs from Vietri's formulation. It is useful to know when we can use
Peacock's approximation because Peacock's approximation is simple to derive the
power-law index. In addition, we focus on how the variance of the energy-gain
factor has an influence on the difference between Vietri's formulation and
Peacock's approximation. The effect of the variance has not been examined well
until now. For demonstration, we consider two cases for the scattering in the
upstream: the large-angle scattering (model A) and the regular deflection by
large-scale magnetic fields (model B). Especially there is no correlation among
the distribution of an energy-gain factor for every step in model A. In this
model, we see the power-law index derived from Peacock's approximation differs
from the one derived from Vietri's formulation when we consider the
mildly-relativistic shock, and the variance of the energy-gain factor affects
this difference. We can use Peacock's approximation for a non-relativistic
shock and a highly-relativistic shock because the effect of the variance is
hidden. In model B, we see the difference of the power-law converging along the
shock velocity.
