Heavy quark distribution function in QCD and the $AC^{2}M^{2}$ model

We show that the phenomenological \ACM ansatz is consistent with QCD through order $1/m_b$ in the description of $B\ra l\bar \nu_l+X_u$ and $B\ra \gamma +X_s$ transitions, including their energy spectra and differential distributions. This suggests a concrete realization for the QCD distribution function, the "Roman" function. On the other hand the endpoint description does not hold for $B\ra l\bar \nu_l+X_c$ decays. The intrinsic limitation of the \ACM model could reveal itself in different values of the effective $b$ quark mass from fits of semileptonic $\Lambda _b$ and $B$ decays.