Our understanding of stellar systems depends on the adopted interpretation of the IMF, phi(m). Unfortunately, there is not a common interpretation of the IMF, which leads to different methodologies and diverging analysis of observational data.We stud
y the correlation between the most massive star that a cluster would host, mmax, and its total mass into stars, M, as an example where different views of the IMF lead to different results. We assume that the IMF is a probability distribution function and analyze the mmax-M correlation within this context. We also examine the meaning of the equation used to derive a theoretical M-char_mmax relationship, N x int[Char_mmax-mup] phi(m) dm = 1 with N the total number of stars in the system, according to different interpretations of the IMF. We find that only a probabilistic interpretation of the IMF, where stellar masses are identically independent distributed random variables, provides a self-consistent result. Neither M nor N, can be used as IMF scaling factors. In addition, Char_mmax is a characteristic maximum stellar mass in the cluster, but not the actual maximum stellar mass. A <M>-Char_mmax correlation is a natural result of a probabilistic interpretation of the IMF; however, the distribution of observational data in the N (or M)-cmmax plane includes a dependence on the distribution of the total number of stars, N (and M), in the system, Phi(N), which is not usually taken into consideration. We conclude that a random sampling IMF is not in contradiction to a possible mmax-M physical law. However, such a law cannot be obtained from IMF algebraic manipulation or included analytically in the IMF functional form. The possible physical information that would be obtained from the N (or M)-mmax correlation is closely linked with the Phi(M) and Phi(N) distributions; hence it depends on the star formation process and the assumed.
We present the underlying relations between colour-magnitude diagrams (CMDs) and synthesis models through the use of stellar luminosity distribution func- tions. CMDs studies make a direct use of the stellar luminosity distribution function while, in
general, synthesis models only use its mean value, even though high-order moments can also be obtained. We show that the mean, high-order moments and in- tegrated luminosity distribution functions of stellar ensembles are related to the stellar luminosity distribution function, within the formalism of probabilistic synthesis mod- els. More details have been yet presented in Cervin ~ o & Luridiana (2006) and references therein. As a direct application of this formalism, we discuss two key issues. First, in- ferences on the upper mass limit of the initial mass function as a function of the total mass of clusters. Second, we apply extreme value theory to show that that the cluster mass obtained from normalising the IMF between mmax and mup does not provide the cluster mass in the case where only one star in this mass range is present, as assumed in the IGIMF theory. It provides instead the cluster mass with a 60% probability to have a star with mass larger than mmax, and we argue that in light of this result the basic formulation ofthe IGIMF theory must be revised.
