Crucial aspects of the initial mass function - II. The inference of total quantities from partial information on a cluster

Context. 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). Aims. 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. Methods. 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. Results. We show that the value of M that would be inferred must be described as a probability distribution ΦM[M;ma,Na, ΦN (N)] that depends on the completeness limit of the data, ma, the number of stars observed down to this limit, Na, and the prior hypothesis made on the distribution of the total number of stars in clusters, ΦN (N).