  We test the parton model approach to inclusive $B \ra \gamma X_s$ and $B \ra
e \bar{\nu} X_u$ processes by calculating a few moments of the distribution
function. The $1/m_Q$ expansion of $b$-quark distribution function is
discussed. We show that Peterson's fragmentation function is useful as the
distribution function of $b$-quark but requires some improvements. By extension
of Peterson's arguments, we obtain the refined form for the distribution
function and with the refined function, we obtain the consistent results with
the HQET.
