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Encyclopedia > Probability distribution

A probability distribution describes the values and probabilities that a random event can take place. The values must cover all of the possible outcomes of the event, while the total probabilities must sum to exactly 1, or 100%. For example, a single coin flip can take values Heads or Tails with a probability of exactly 1/2 for each; these two values and two probabilities make up the probability distribution of the single coin flipping event. This distribution is called a discrete distribution because there are a countable number of discrete outcomes with positive probabilities. Image File history File links Question_book-3. ... The word probability derives from the Latin probare (to prove, or to test). ... In probability theory, an event is a set of outcomes (a subset of the sample space) to which a probability is assigned. ... Addition is one of the basic operations of arithmetic. ... In mathematics, a probability distribution is called discrete, if it is fully characterized by a probability mass function. ... In mathematics the term countable set is used to describe the size of a set, e. ...

A continuous distribution describes events over a continuous range, where the probability of a specific outcome is zero. For example, a dart thrown at a dartboard has essentially zero probability of landing at a specific point, since a point is vanishingly small, but it has some probability of landing within a given area. The probability of landing within the small area of the bullseye would (hopefully) be greater than landing on an equivalent area elsewhere on the board. A smooth function that describes the probability of landing anywhere on the dartboard is the probability distribution of the dart throwing event. The integral of the probability density function (pdf) over the entire area of the dartboard (and, perhaps, the wall surrounding it) must be equal to 1, since each dart must land somewhere. By one convention, a probability distribution is called continuous if its cumulative distribution function is continuous. ... Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. ... This article is about the concept of integrals in calculus. ... In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ...

In probability theory, every random variable may be attributed to a function defined on a state space equipped with a probability distribution that assigns a probability to every subset (more precisely every measurable subset) of its state space in such a way that the probability axioms are satisfied. That is, probability distributions are probability measures defined over a state space instead of the sample space. A random variable then defines a probability measure on the sample space by assigning a subset of the sample space the probability of its inverse image in the state space. In other words the probability distribution of a random variable is the push forward measure of the probability distribution on the state space. Probability theory is the branch of mathematics concerned with analysis of random phenomena. ... In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ... Probability is the likelihood that something is the case or will happen. ... â€œSupersetâ€ redirects here. ... In mathematics, the definition of the probability space is the foundation of probability theory. ... In probability theory, the probability P of some event E, denoted , is defined in such a way that P satisfies the Kolmogorov axioms. ... In mathematics, a probability space is a set S, together with a σ-algebra X on S and a measure P on that σ-algebra such that P(S) = 1. ... In probability theory, the sample space or universal sample space, often denoted S, Î© or U (for universe), of an experiment or random trial is the set of all possible outcomes. ... In mathematics, a pushforward measure (also push forward or push-forward) is obtained by transferring (pushing forward) a measure from one measurable space to another using a measurable function. ...

### Probability distributions of real-valued random variables

Because a probability distribution Pr on the real line is determined by the probability of being in a half-open interval Pr(ab], the probability distribution of a real-valued random variable X is completely characterized by its cumulative distribution function: In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than...

#### Discrete probability distribution

A probability distribution is called discrete if its cumulative distribution function only increases in jumps. In mathematics, a probability distribution is called discrete, if it is fully characterized by a probability mass function. ...

The set of all values that a discrete random variable can assume with non-zero probability is either finite or countably infinite because the sum of uncountably many positive real numbers (which is the smallest upper bound of the set of all finite partial sums) always diverges to infinity. Typically, the set of possible values is topologically discrete in the sense that all its points are isolated points. But, there are discrete random variables for which this countable set is dense on the real line. In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... In mathematics the term countable set is used to describe the size of a set, e. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In topology, a branch of mathematics, a point x of a set S is called an isolated point, if there exists a neighborhood of x not containing other points of S. In particular, in a Euclidean space (or in a metric space), x is an isolated point of S, if... In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if, intuitively, any point in X can be well-approximated by points in A. Formally, A is dense in X if for any point x in X, any neighborhood of...

Discrete distributions are characterized by a probability mass function, p such that In probability theory, a probability mass function (abbreviated pmf) gives the probability that a discrete random variable is exactly equal to some value. ...

#### Continuous probability distribution

By one convention, a probability distribution is called continuous if its cumulative distribution function is continuous, which means that it belongs to a random variable X for which Pr[ X = x ] = 0 for all x in R. By one convention, a probability distribution is called continuous if its cumulative distribution function is continuous. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...

Another convention reserves the term continuous probability distribution for absolutely continuous distributions. These distributions can be characterized by a probability density function: a non-negative Lebesgue integrable function f defined on the real numbers such that // Absolute continuity of real functions In mathematics, a real-valued function f of a real variable is absolutely continuous on a specified finite or infinite interval if for every positive number Îµ, no matter how small, there is a positive number Î´ small enough so that whenever a sequence of pairwise disjoint... In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ... The integral of a positive function can be interpreted as the area under a curve. ...

Discrete distributions and some continuous distributions (like the devil's staircase) do not admit such a density. In mathematics, a devils staircase is any function f(x) defined on the interval [a, b] that has the following properties: f(x) is continuous on [a, b]. there exists a set N of measure 0 such that for all x outside of N the derivative fâ€²(x) exists...

### Terminology

The support of a distribution is the smallest closed set whose complement has probability zero.

The probability distribution of the sum of two independent random variables is the convolution of each of their distributions. In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. ...

The probability distribution of the difference of two random variables is the cross-correlation of each of their distributions. In statistics, the term cross-correlation is sometimes used to refer to the covariance cov(X, Y) between two random vectors X and Y, in order to distinguish that concept from the covariance of a random vector X, which is understood to be the matrix of covariances between the scalar...

A discrete random variable is a random variable whose probability distribution is discrete. Similarly, a continuous random variable is a random variable whose probability distribution is continuous.

## List of important probability distributions

Certain random variables occur very often in probability theory, in some cases due to their application to many natural and physical processes, and in some cases due to theoretical reasons such as the central limit theorem, the Poisson limit theorem, or properties such as memorylessness or other characterizations. Their distributions therefore have gained special importance in probability theory. Image File history File links No higher resolution available. ... A central limit theorem is any of a set of weak-convergence results in probability theory. ... In probability theory, memorylessness is a property of certain probability distributions: the exponential distributions and the geometric distributions. ... In the jargon of mathematics, the statement that Property P characterizes object X means, not simply that X has property P, but that X is the only thing that has property P. It is also common to find statements such as Property Q characterises Y up to isomorphism. The first...

### Discrete distributions

#### With finite support

• The Bernoulli distribution, which takes value 1 with probability p and value 0 with probability q = 1 − p.
• The Rademacher distribution, which takes value 1 with probability 1/2 and value −1 with probability 1/2.
• The binomial distribution describes the number of successes in a series of independent Yes/No experiments.
• The degenerate distribution at x0, where X is certain to take the value x0. This does not look random, but it satisfies the definition of random variable. It is useful because it puts deterministic variables and random variables in the same formalism.
• The discrete uniform distribution, where all elements of a finite set are equally likely. This is supposed to be the distribution of a balanced coin, an unbiased die, a casino roulette or a well-shuffled deck. Also, one can use measurements of quantum states to generate uniform random variables. All these are "physical" or "mechanical" devices, subject to design flaws or perturbations, so the uniform distribution is only an approximation of their behaviour. In digital computers, pseudo-random number generators are used to produce a statistically random discrete uniform distribution.
• The hypergeometric distribution, which describes the number of successes in the first m of a series of n Yes/No experiments, if the total number of successes is known.
• Zipf's law or the Zipf distribution. A discrete power-law distribution, the most famous example of which is the description of the frequency of words in the English language.
• The Zipf-Mandelbrot law is a discrete power law distribution which is a generalization of the Zipf distribution.

In probability theory and statistics, the Bernoulli distribution, named after Swiss scientist Jakob Bernoulli, is a discrete probability distribution, which takes value 1 with success probability and value 0 with failure probability . ... In probability theory and statistics, the Rademacher distribution is a discrete probability distribution. ... In probability theory and statistics, the binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. ... In mathematics, a degenerate distribution is the probability distribution of a random variable which always has the same value. ... In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ... In probability theory and statistics, the discrete uniform distribution is a discrete probability distribution that can be characterized by saying that all values of a finite set of possible values are equally probable. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... A Pseudorandom number sequence is a sequence of numbers that has been computed by some defined arithmetic process but is effectively a random number sequence for the purpose for which it is required. ... Random redirects here. ... // In probability theory and statistics, the hypergeometric distribution is a discrete probability distribution that describes the number of successes in a sequence of n draws from a finite population without replacement. ... Originally, Zipfs law stated that, in a corpus of natural language utterances, the frequency of any word is roughly inversely proportional to its rank in the frequency table. ... In probability theory and statistics, the Zipf-Mandelbrot law is a discrete probability distribution. ... Originally the term Zipfs law meant the observation of Harvard linguist George Kingsley Zipf (SAMPA: [zIf]) that the frequency of use of the nth-most-frequently-used word in any natural language is approximately inversely proportional to n. ...

#### With infinite support

• The Boltzmann distribution, a discrete distribution important in statistical physics which describes the probabilities of the various discrete energy levels of a system in thermal equilibrium. It has a continuous analogue. Special cases include:
• The Gibbs distribution
• The Maxwell-Boltzmann distribution
• The Bose-Einstein distribution
• The Fermi-Dirac distribution
• The geometric distribution, a discrete distribution which describes the number of attempts needed to get the first success in a series of independent Yes/No experiments.
• The Skellam distribution, the distribution of the difference between two independent Poisson-distributed random variables
• The Yule-Simon distribution
• The zeta distribution has uses in applied statistics and statistical mechanics, and perhaps may be of interest to number theorists. It is the Zipf distribution for an infinite number of elements.

### Continuous distributions

#### Supported on a bounded interval

• The Beta distribution on [0,1], of which the uniform distribution is a special case, and which is useful in estimating success probabilities.

#### Supported on semi-infinite intervals, usually [0,∞)

• The exponential distribution, which describes the time between consecutive rare random events in a process with no memory.
• The F-distribution, which is the distribution of the ratio of two (normalized) chi-square distributed random variables, used in the analysis of variance. (Called the beta prime distribution when it is the ratio of two chi-square variates which are not normalized by dividing them by their numbers of degrees of freedom.)
• The noncentral F-distribution
• The Gamma distribution, which describes the time until n consecutive rare random events occur in a process with no memory.
• The folded normal distribution
• The half-normal distribution
• The inverse Gaussian distribution, also known as the Wald distribution
• The Lévy distribution
• The log-logistic distribution
• The log-normal distribution, describing variables which can be modelled as the product of many small independent positive variables.

### Joint distributions

For any set of independent random variables the probability density function of their joint distribution is the product of their individual density functions. In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ... Given two random variables X and Y, the joint probability distribution of X and Y is the probability distribution of X and Y together. ...

#### Two or more random variables on the same sample space

Several images of the probability density of the Dirichlet distribution when K=3 for various parameter vectors Î±. Clockwise from top left: Î±=(6, 2, 2), (3, 7, 5), (6, 2, 6), (2, 3, 4). ... In probability theory and statistics, the beta distribution is a continuous probability distribution with the probability density function (pdf) defined on the interval [0, 1]: where Î± and Î² are parameters that must be greater than zero and B is the beta function. ... In population genetics, Ewenss sampling formula, introduced by Warren Ewens, states that under certain conditions (specified below), if a random sample of n gametes is taken from a population and classified according to the gene at a particular locus then the probability that there are a1 alleles represented once... In mathematics, a partition of a positive integer n is a way of writing n as a sum of positive integers. ... Population genetics is the study of the distribution of and change in allele frequencies under the influence of the four evolutionary forces: natural selection, genetic drift, mutation, and migration. ... The Balding-Nichols model is a statistical description of the allele frequencies in the components of a sub-divided population. ... In probability theory, the multinomial distribution is a generalization of the binomial distribution. ... In probability theory and statistics, the binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. ... In probability theory and statistics, a multivariate normal distribution, also sometimes called a multivariate Gaussian distribution, is a specific probability distribution, which can be thought of as a generalization to higher dimensions of the one-dimensional normal distribution (also called a Gaussian distribution). ... The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ...

#### Matrix-valued distributions

In statistics, the Wishart distribution, named in honor of John Wishart, is any of a family of probability distributions for nonnegative-definite matrix-valued random variables (random matrices). These distributions are of great importance in the estimation of covariance matrices in multivariate statistics. ... The matrix normal distribution is a probability distribution that is a generalization of the normal distribution. ... In statistics, Hotellings T-square statistic, named for Harold Hotelling, is a generalization of Students t statistic that is used in multivariate hypothesis testing. ...

### Miscellaneous distributions

The Cantor distribution is the probability distribution whose cumulative distribution function is the Cantor function. ... A phase-type distribution is a probability distribution that results from a system of one or more inter-related Poisson processes occurring in sequence, or phases. ... A truncated distribution is a conditional distribution that conditions on the random variable in question. ...

Results from FactBites:

 Probability distribution (0 words) The Probability distribution of the sum of two random variables is the convolution of each of their distributions. The Probability distribution of the difference of two random variables is the cross-correlation of each of their distributions. The triangular distribution on [''a'', b], a special case of which is the distribution of the sum of two uniformly distributed random variables (the convolution of two uniform distributions).
 VBA11 - Multivariate Standard Normal Probability Distribution (317 words) The procedure for generating random numbers from a multivariate distribution is described in the 4 steps of the example shown later. The probability computed in this program is based on the area under the probability distribution from negative infinity to z. For example, the standard deviation and the mean from a standard normal probability distribution should be 1 and 0, respectively.
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