Probability Distribution

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

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



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]\{\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.


Number of Heads Probability
0 0.25
1 0.5
2 0.25


  • (Wikipedia, 2015) ⇒
    • 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.
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.