Probability Distribution

A Probability Distribution is a table or function that describes the probability of occurrence of each possible outcome in a statistical experiment.

• AKA: Probability Distribution Table, Probability Distribution Function.
• Context:
  • It can be a normalized frequency distribution table or function.
  • It can be a probability measure function of a random variable.
  • It can range from being a Probability Mass Function, to being a Probability Density Function, to being a Cumulative Distribution Function.
  • It can range from Univariate Probability Distribution Function to being a Multivariate Probability Distribution Function.
  • It can (often) be a member of a Probability Function Set (such as defined by a probability distribution family).
• Example(s):
  • Probability distribution table of all possible outcomes of the statistical experiment of drawing 2 balls at random in succession without replacement from a urn containing 4 white balls and 6 black balls

Possible Outcomes	Number of White Balls Drawn	Number of Black Balls Drawn	Probability
WW	2	0	2/15
WB	1	1	4/15
BW	1	1	4/15
BB	0	2	1/3

• Probability Distribution Functions:

• Uniform Distribution
• Binomial Distribution
• Log-Normal Distribution
• Bernoulli Distribution
• Geometric Distribution
• Hypergeometric Distribution.

• Counter-Example(s):
  • Frequency Distribution.
  • Complex Function.
• See: Probability Measure, Statistical Model Family, Sample Space, Probability Space.

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References

2017a

• (Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/Probability_distribution
  • In probability theory and statistics, a probability distribution is a mathematical function that, stated in simple terms, can be thought of as providing the probability of occurrence of different possible outcomes in an experiment. For instance, if the random variable X is used to denote the outcome of a coin toss ('the experiment'), then the probability distribution of X would take the value 0.5 for [math]\displaystyle{ \text{X=Heads} }[/math], and 0.5 for [math]\displaystyle{ \text{X=Tails} }[/math].

In more technical terms, the probability distribution is a description of a random phenomenon in terms of the probabilities of events. Examples of random phenomena can include the results of an experiment or survey. A probability distribution is defined in terms of an underlying sample space, which is the set of all possible outcomes of the random phenomenon being observed. The sample space may be the set of real numbers or a higher-dimensional vector space, or it may be a list of non-numerical values; for example, the sample space of a coin flip would be [math]\displaystyle{ \{\text{Heads},\text{Tails}\} }[/math].

Probability distributions are generally divided into two classes. A discrete probability distribution (applicable to the scenario where the set of possible outcomes is discrete, such as a coin toss or a roll of dice) can be encoded by a discrete list of the probabilities of the outcomes, known as a probability mass function. On the other hand, a continuous probability distribution (applicable to the scenarios where the set of possible outcomes can take on values in a continuous range (e.g., real numbers), such as the temperature on a given day) is typically described by probability density functions (with the probability of any individual outcome actually being 0). The normal distribution represents a commonly encountered continuous probability distribution. More complex experiments, such as those involving stochastic processes defined in continuous time, may demand the use of more general probability measures.

A probability distribution whose sample space is the set of real numbers is called univariate, while a distribution whose sample space is a vector space is called multivariate. A univariate distribution gives the probabilities of a single random variable taking on various alternative values; a multivariate distribution (a joint probability distribution) gives the probabilities of a random vector — a list of two or more random variables — taking on various combinations of values. Important and commonly encountered univariate probability distributions include the binomial distribution, the hypergeometric distribution, and the normal distribution. The multivariate normal distribution is a commonly encountered multivariate distribution.

2017b

• (Stattrek, 2017) ⇒ http://stattrek.com/probability-distributions/probability-distribution.aspx
  • A probability distribution is a table or an equation that links each outcome of a statistical experiment with its probability of occurrence. Consider the coin flip experiment described above. The table below, which associates each outcome with its probability, is an example of a probability distribution.

Number of Heads	Probability
0	0.25
1	0.5
2	0.25

2017c

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Probability_distribution#Terminology
  • As probability theory is used in quite diverse applications, terminology is not uniform and sometimes confusing. The following terms are used for non-cumulative probability distribution functions:
    • Distribution, Frequency distribution: is a table that displays the frequency of various outcomes in a sample.
    • Probability distribution: is a table that displays the probabilities of various outcomes in a sample. Could be called a "normalized frequency distribution table", where all occurrences of outcomes sum to 1.
    • Distribution function: is a functional form of frequency distribution table.
    • Probability distribution function: is a functional form of probability distribution table. Could be called a "normalized frequency distribution fuction", where area under the graph equals to 1.

Finally,

• Probability mass, Probability mass function, p.m.f., Discrete probability distribution function: for discrete random variables.
• Categorical distribution: for discrete random variables with a finite set of values.
• Probability density, Probability density function, p.d.f., Continuous probability distribution function: most often reserved for continuous random variables.

The following terms are somewhat ambiguous as they can refer to non-cumulative or cumulative distributions, depending on authors' preferences:

• Probability distribution function: continuous or discrete, non-cumulative or cumulative.
• Probability function: even more ambiguous, can mean any of the above or other things.

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