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
  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 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. 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.
