Massfunktion

A massfunktion, in statistics and probability theory, is a function that assigns probabilities to the possible discrete outcomes of a random variable. If X is a discrete random variable with possible values in a set S, the massfunktion p_X satisfies p_X(x) = P(X = x) for x in S, with p_X(x) ≥ 0 and the sum over all x in S equal to 1. It is distinct from a probability density function, which is used for continuous variables, and it completely specifies the distribution of X. The cumulative distribution function F_X(x) = P(X ≤ x) is obtained by summing p_X(y) over y ≤ x.

Common examples include a fair die, where p(1)=...=p(6)=1/6, and a Bernoulli variable with p(1)=p and p(0)=1−p. The Poisson distribution is a widely used massfunktion on the nonnegative integers, defined by p(k)=e^{−λ} λ^k / k! for k=0,1,2,...

In astronomy and astrophysics, the term Massfunktion is used to describe the mass distribution of a population of objects, such as stars or galaxies. The stellar initial mass function (IMF) specifies the number of stars formed per unit mass interval and is often expressed as dN/dM ∝ M^{−α} over a mass range, with parameterizations such as the Salpeter or Kroupa forms. Such massfunktion descriptions are used to derive total mass, light, or chemical yield for stellar populations.