  In this paper, we give a general discussion on the calculation of the
statistical distribution from a given operator relation of creation,
annihilation, and number operators. Our result shows that as long as the
relation between the number operator and the creation and annihilation
operators can be expressed as $a^{\dagger}b=\Lambda\left(N\right) $ or
$N=\Lambda^{-1} \left( a^{\dagger}b\right)$, where $N$, $a^{\dagger}$, and $b$
denote the number, creation, and annihilation operators, i.e., $N$ is a
function of quadratic product of the creation and annihilation operators, the
corresponding statistical distribution is the Gentile distribution, a
statistical distribution in which the maximum occupation number is an arbitrary
integer. As examples, we discuss the statistical distributions corresponding to
various operator relations. In particular, besides Bose-Einstein and
Fermi-Dirac cases, we discuss the statistical distributions for various schemes
of intermediate statistics, especially various $q$-deformation schemes. Our
result shows that the statistical distributions corresponding to various
$q$-deformation schemes are various Gentile distributions with different
maximum occupation numbers which are determined by the deformation parameter
$q$. This result shows that the results given in much literature on the
$q$-deformation distribution are inaccurate or incomplete.
