IMF slope derived from a pure probabilistic model

The stellar initial mass function is of great significance for the study of star formation and galactic structure. Observations indicate that the IMF follows a power-law form. This work derived that when the expected number of stars formed from a spherical molecular cloud is much greater than 1, there is a relationship between the slope $\alpha$ of the IMF and $r^n$ in the radius-density relation of spherically symmetric gas clouds, given by $\alpha = 3/(n+3)$ ($\Gamma_{\mathrm {IMF}} = n/(n+3)$). This conclusion is close to the results of numerical simulations and observations, but it is derived from a pure probabilistic model, which may have underlying reasons worth pondering.