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A probability mass function (PMF) is a function on a discrete random variable that gives the relative likelihood of the variable taking a given value. (The phrase's use of the term "mass" is at-best by analogy and does not refer to physical mass: see below.) A function that produces values that are some multiple of those of such a PMF (i.e., such that the ratio between any two of its values are the same as those of the PMF) is termed an unnormalized PMF and the equivalent normalized PMF can be derived by incorporating an additional constant factor, the normalizing constant aka normalization constant, which is the reciprocal of the sum of the unnormalized PMF's results.
A discrete random variable, as opposed to a continuous random variable, is a random variable which takes only discrete values: a variable that can be any real number is the latter, whereas a variable that can be any integer, or a variable that can be only the integers 1 through 6 is the former. Simple examples if PMFs (of discrete random variable X):
An example of a PMF over an infinite number of values is:
In this latter instance, a series of f(X)'s results for X = 1,2,3,... has a limit of 1.
Note that in astrophysics the phrase mass function is used for some functions (including some distribution functions) that have to do with physical mass, and do not fit the probability term probability mass function. One example is the mass function used for binary stars, and other examples are models of the mass distribution of astronomical objects, such as the initial mass function and conditional stellar mass function, which might be classified as (possibly unnormalized) probability density functions.