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Our understanding of stellar systems depends on the adopted interpretation ofthe IMF, phi(m). Unfortunately, there is not a common interpretation of theIMF, which leads to different methodologies and diverging analysis ofobservational data.We study the correlation between the most massive star thata cluster would host, mmax, and its total mass into stars, M, as an examplewhere different views of the IMF lead to different results. We assume that theIMF is a probability distribution function and analyze the mmax-M correlationwithin this context. We also examine the meaning of the equation used to derivea theoretical M-char_mmax relationship, N x int[Char_mmax-mup] phi(m) dm = 1with N the total number of stars in the system, according to differentinterpretations of the IMF. We find that only a probabilistic interpretation ofthe IMF, where stellar masses are identically independent distributed randomvariables, provides a self-consistent result. Neither M nor N, can be used asIMF scaling factors. In addition, Char_mmax is a characteristic maximum stellarmass in the cluster, but not the actual maximum stellar mass. A <M>-Char_mmaxcorrelation is a natural result of a probabilistic interpretation of the IMF;however, the distribution of observational data in the N (or M)-cmmax planeincludes a dependence on the distribution of the total number of stars, N (andM), in the system, Phi(N), which is not usually taken into consideration. Weconclude that a random sampling IMF is not in contradiction to a possiblemmax-M physical law. However, such a law cannot be obtained from IMF algebraicmanipulation or included analytically in the IMF functional form. The possiblephysical information that would be obtained from the N (or M)-mmax correlationis closely linked with the Phi(M) and Phi(N) distributions; hence it depends onthe star formation process and the assumed.

astronomy and astrophysics

Crucial aspects of the initial mass function (I): The statistical  correlation between the total mass of an ensemble of stars and its most  massive star