## Crucial aspects of the initial mass function

### II. The inference of total quantities from partial information on a cluster

^{1}
Instituto de Astrofísica de Andalucía (IAA-CSIC),
Glorieta de la Astronomía s/n,
18008
Granada,
Spain

e-mail: mcs@iaa.es

^{2}
Instituto de Astrofísica de Canarias, c/vía Láctea s/n, 38205
La Laguna, Tenerife,
Spain

^{3}
Instituto de Astronomía, Universidad Académica en Ensenada,
Universidad Nacional Autónoma de México, Ensenada BC, 22860
Mexico,
Mexico

^{4}
European Southern Observatory, Casilla 19001, Santiago 19, Chile

^{5}
Max Planck Institut für Astronomie, Königstuhl 17, 69117
Heidelberg,
Germany

^{6}
Departamento de Astrofísica, Universidad de La Laguna
(ULL), 38205, La
Laguna, Tenerife,
Spain

^{7}
S. D. Astronomía y Geodesia, Fac. CC. Matemáticas, Universidad
Complutense de Madrid, 28040
Madrid,
Spain

Received:
17
December
2012

Accepted:
21
February
2013

*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 Φ_{ℳ}[ℳ;*m*_{a},*N*_{a},Φ_{N}(*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,
Φ_{N}(*N*).

Key words: stars: statistics / galaxies: stellar content / methods: data analysis

*© ESO, 2013*