### probability mass function

**(PMF)**
(shows probability of taking one of a number of discrete values)

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 numbers having the same ratios
between each other but do not sum to one
can be considered an **unnormalized PMF**
and the equivalent **normalized PMF** can be derived by
incorporating an additional constant factor (**normalizing constant** aka
**normalization constant**),
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 numbers 0 through 6 are the former.
Simple examples if PMFs (of discrete random variable X):

- for X either heads or tails, f(X) = 1/2 (models a coin flip).
- for X being an integer 1 through 6, f(X) = 1/6 (models the roll of a die).

An example of a PMF over an infinite number of values is:

- for X being a non-negative integer, f(X) = 1/2
^{X}, i.e., f(1) = 1/2, f(2) = 1/4, f(3) = 1/8, ...

In this case, a series of f(X)'s results has a limit of 1.

Note that in astrophysics the phrase **mass function** is used for
functions that have to do with physical mass, and not related
to a *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 (or are similar to) probability density functions.

(*mathematics,statistics,function*)
**Further reading:**

http://en.wikipedia.org/wiki/Probability_mass_function

**Referenced by pages:**

Akaike information criterion (AIC)

Bayesian statistics

conditional stellar mass function (CSMF)

dense core mass function (DCMF)

halo mass function

initial mass function (IMF)

mass function

Navarro-Frenk-White profile (NFW profile)

probability density function (PDF)

Index