Density probability distribution functions (PDFs) for turbulent self-gravitating clouds should be convolutions of the local log-normal PDF, which depends on the local average density ρave and Mach number, and the PDFs for ρave and, which depend on the overall cloud structure. When self-gravity drives a cloud to increased central density, the total PDF develops an extended tail. If there is a critical density or column density for star formation, then the fraction of the local mass exceeding this threshold becomes higher near the cloud center. These elements of cloud structure should be in place before significant star formation begins. Then the efficiency is high so that bound clusters form rapidly, and the stellar initial mass function (IMF) has an imprint in the gas before destructive radiation from young stars can erase it. The IMF could arise from a power-law distribution of mass for cloud structure. These structures should form stars down to the thermal Jeans mass MJ at each density in excess of a threshold. The high-density tail of the PDF, combined with additional fragmentation in each star-forming core, extends the IMF into the brown dwarf regime. The core fragmentation process is distinct from the cloud structuring process and introduces an independent core fragmentation mass function (CFMF). The CFMF would show up primarily below the IMF peak. © 2011. The American Astronomical Society. All rights reserved.