Crucial aspects of the initial mass function (II): The inference of
  total quantities from partial information on a cluster
  In a probabilistic framework of the interpretation of the initial mass
function (IMF), the IMF cannot be arbitrarily normalized to the total mass, M,
or number of stars, N, of the system. Hence, the inference of M and N when
partial information about the studied system is available must be revised.
(i.e., the contribution to the total quantity cannot be obtained by simple
algebraic manipulations of the IMF). We study how to include constraints in the
IMF to make inferences about different quantities characterizing stellar
systems. It is expected that including any particular piece of information
about a system would constrain the range of possible solutions. However,
different pieces of information might be irrelevant depending on the quantity
to be inferred. In this work we want to characterize the relevance of the
priors in the possible inferences. Assuming that the IMF is a probability
distribution function, we derive the sampling distributions of M and N of the
system constrained to different types of information available. We show that
the value of M that would be inferred must be described as a probability
distribution Phi[M; m_a, N_a, Phi(N)] that depends on the completeness limit of
the data, m_a, the number of stars observed down to this limit, N_a, and the
prior hypothesis made on the distribution of the total number of stars in
clusters, Phi(N).
