Discrete Probability Function

A discrete probability function is a probability distribution function that represents a discrete random variable.

• AKA: [math]\displaystyle{ \operatorname{PMF}_X(x) }[/math], [math]\displaystyle{ \operatorname{PMF}(X,x) }[/math], Probability Mass.
• Context:
  • inputs:
    • a Random Experiment Identifier (to a discrete random experiment), [math]\displaystyle{ X }[/math].
    • a Random Experiment Outcome Identifier, [math]\displaystyle{ x }[/math].
    • Optional Input: Parameters.
  • range: a Probability Value.
  • It can be a member of a Discrete Probability Distribution Family.
  • It can range from being an Abstract Discrete Probability Function to being a Discrete Probability Structure (such as a discrete probability software function).
  • It can range from being a Uniform Discrete Probability Function to being a Non-Uniform Discrete Probability Function.
  • It can range from being a Binomial Probability Function to being a Multinomial Probability Function.
  • …
• Example(s):
  • [math]\displaystyle{ p(\text{Heads})=0.5; p(\text{Tails})=0.5 }[/math] for a Coin-Toss Experiment.
  • a Geometric Probability Function, from a geometric probability distribution.
  • a Negative Binomial Probability Function, from a negative binomial probability distribution.
  • a Beta-Binomial Probability Function, from a beta-binomial probability function.
  • a Conditional Probability Mass Function.
  • an N-gram Model.
  • …
• Counter-Example(s):
  • Continuous Probability Function.
• See: Poisson Distribution, Multivariate Random Variable.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/probability_mass_function Retrieved:2015-6-2.
  • In probability theory and statistics, a probability mass function (pmf) is a function that gives the probability that a discrete random variable is exactly equal to some value. The probability mass function is often the primary means of defining a discrete probability distribution, and such functions exist for either scalar or multivariate random variables whose domain is discrete. A probability mass function differs from a probability density function (pdf) in that the latter is associated with continuous rather than discrete random variables; the values of the latter are not probabilities as such: a pdf must be integrated over an interval to yield a probability.

• http://en.wikibooks.org/wiki/Statistics/Distributions/Discrete#Probability_Mass_Function
  • A discrete random variable has a probability mass function that describes how likely the random variable is to be at a certain point. The probability mass function must have a total of 1, and sums to the cdf. The pmf is represented by the lowercase f.

2009

• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Discrete_probability_distribution
  • Discrete probability distributions arise in the mathematical description of probabilistic and statistical problems in which the values that might be observed are restricted to being within a pre-defined list of possible values. This list has either a finite number of members, or at most is countable.

2006

• (Dubnicka, 2006c) ⇒ Suzanne R. Dubnicka. (2006). “Random Variables - STAT 510: Handout 3." Kansas State University, Introduction to Probability and Statistics I, STAT 510 - Fall 2006.
  • TERMINOLOGY : The function pX(x) = P(X = x) is called the probability mass function (pmf) for the discrete random variable X.
  • FACTS: The pmf pX(x) for a discrete random variable X consists of two parts:
    • (a) SX, the support set of X, and
    • (b) a probability assignment P(X = x), for all x 2
  • The pmf of a discrete random variable and the pdf of a continuous random variable provides complete information about the probabilistic properties of a random variable. However, it is sometimes useful to employ summary measures. The most basic summary measure is the expectation or mean of a random variable X, denoted E(X), which can be thought of as an “average” value of a random variable.

2000

• (Valpola, 2000) ⇒ Harri Valpola. (2000). “Bayesian Ensemble Learning for Nonlinear Factor Analysis." PhD Dissertation, Helsinki University of Technology.
  • QUOTE: probability mass: In analogy to physical mass and density, ordinary probability can be called probability mass in order to distinguish it from probability density.

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