The plasma membrane of eukaryotic cells is known to be compositionally asymmetric. Certain phospholipids, such as sphingomyelin and phosphatidylcholine species, are predominantly localized in the outer leaflet, while phosphatidylethanolamine and phosphatidylserine species primarily reside in the inner leaflet. While phospholipid asymmetry between the membrane leaflets is well established, there is no consensus about cholesterol distribution between the two leaflets. We have performed a systematic study, via molecular simulations, of how the spatial distribution of cholesterol molecules in different "asymmetric" lipid bilayers are affected by the lipids' backbone, head-type, unsaturation, and chain-length by considering an asymmetric bilayer mimicking the plasma membrane lipids of red blood cells, as well as seventeen other asymmetric bilayers comprising of different lipid types. Our results reveal that the distribution of cholesterol in the leaflets is solely a function of the extent of ordering of the lipids within the leaflets. The ratio of the amount of cholesterol matches the ratio of lipid order in the two leaflets, thus providing a quantitative relationship between the two. These results are understood by the observation that asymmetric bilayers with equimolar amount of lipids in the two leaflets develop tensile and compressive stresses due to differences in the extent of lipid order. These stresses are alleviated by the transfer of cholesterol from the leaflet in compressive stress to the one in tensile stress. These findings are important in understanding the biology of the cell membrane, especially with regard to the composition of the membrane leaflets.