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

Context. Our understanding of stellar systems depends on the adopted interpretation of the initial mass function, IMF phi(m). Unfortunately, there is not a common interpretation of the IMF, which leads to different methodologies and diverging analysis of observational data. Aims. We study 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. Methods. We assume that the IMF is a probability distribution function and analyze the m(max) - Mcorrelation within this context. We also examine the meaning of the equation used to derive a theoretical M - (m) over cap max relationship, N x integral(max)((m) over cap) phi(m)dm = 1 with N the total number of stars in the system, according to different interpretations of the IMF. Results. 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 the total number of stars in the cluster, N, can be used as IMF scaling factors. In addition, (m) over cap (max) is a characteristic maximum stellar mass in the cluster, but not the actual maximum stellar mass. A (M) -(m) over cap (max) correlation is a natural result of a probabilistic interpretation of the IMF; however, the distribution of observational data in the N (or M) - m(max) plane includes a dependence on the distribution of the total number of stars, N (and M), in the system, FN(N), which is not usually taken into consideration. Conclusions. We conclude that a random sampling IMF is not in contradiction to a possible m(max) - 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)- m(max) correlation is closely linked with the Phi(M)(M) and Phi(N)(N) distributions; hence it depends on the star formation process and the assumed definition of stellar cluster.