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Encyclopedia > Statistical distribution

In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. In technical terms, a probability distribution is a probability measure whose domain is the Borel algebra on the reals. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... The word probability derives from the Latin probare (to prove, or to test). ... The probability of some event (denoted ) is defined with respect to a universe or sample space of all possible elementary events in such a way that must satisfy 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 mathematics, the Borel algebra (or Borel σ-algebra) on a topological space X is a σ-algebra of subsets of X associated to the topology of X. In the mathematics literature, there are at least two inequivalent definitions of this σ-algebra: The minimal σ-algebra containing the open sets. ...


A probability distribution is a special case of the more general notion of a probability measure, which is a function that assigns probabilities satisfying the Kolmogorov axioms to the measurable sets of a measurable space. 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. ... The probability of some event (denoted ) is defined with respect to a universe or sample space of all possible elementary events in such a way that must satisfy the Kolmogorov axioms. ... In mathematics, a σ-algebra (or σ-field) X over a set S is a family of subsets of S which is closed under countable set operations; σ-algebras are mainly used in order to define measures on S. The concept is important in mathematical analysis and probability theory. ...


Every random variable gives rise to a probability distribution, and this distribution contains most of the important information about the variable. If X is a random variable, the corresponding probability distribution assigns to the interval [a, b] the probability Pr[aXb], i.e. the probability that the variable X will take a value in the interval [a, b]. A random variable can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result. ...


The probability distribution of the variable X can be uniquely described by its cumulative distribution function F(x), which is defined by 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 variable X takes on a value less than or...

F(x) = Prleft[ X le x right]

for any x in R.


A distribution is called discrete if its cumulative distribution function consists of a sequence of finite jumps, which means that it belongs to a discrete random variable X: a variable which can only attain values from a certain finite or countable set. A 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. In mathematics, a random variable is discrete if its probability distribution is discrete; a discrete probability distribution is one that is fully characterized by a probability mass function. ... In mathematics the term countable set is used to describe the size of a set, e. ... In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ...


The so-called absolutely continuous distributions can be expressed by a probability density function: a non-negative Lebesgue integrable function f defined on the reals such that In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ... The integral can be interpreted as the area under a curve. ...

Pr left[ a le X le b right] = int_a^b f(x),dx

for all a and b. That discrete distributions do not admit such a density is unsurprising, but there are continuous distributions like the devil's staircase that also do not admit 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 f′(x) exists...

  • 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.
  • The probability distribution of the difference of two random variables is the cross-correlation of each of their distributions.

Contents

For the computer science usage see convolution (computer science) . 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... 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...


List of important probability distributions

Several probability distributions are so important in theory or applications that they have been given specific names:


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, which 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. This 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 produced a statistically random discrete uniform distribution.
  • The hypergeometric distribution, which describes the number of successes in the first m of a series of n independent 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 James Bernoulli, is a discrete probability distribution, which takes value 1 with success probability and value 0 with failure probability . ... See binomial (disambiguation) for a list of other topics using that name. ... In mathematics, a degenerate distribution is the probability distribution of a random variable which always has the same value. ... A random variable can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result. ... 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. ... It has been suggested that random number be merged into this article or section. ... 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. ... This article may be too technical for most readers to understand. ... 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.

The Maxwell-Boltzmann distribution is a probability distribution with applications in physics and chemistry. ... Statistical physics, one of the fundamental theories of physics, uses methods of statistics in solving physical problems. ... In thermodynamics, a thermodynamic system is in thermodynamic equilibrium if its energy distribution equals a Maxwell-Boltzmann-distribution. ... In statistical mechanics, the Gibbs algorithm, first introduced by J. Willard Gibbs in 1878, is the injunction to choose a statistical ensemble (probability distribution) for the unknown microscopic state of a thermodynamic system by minimising the average log probability subject to the probability distribution satisfying a set of constraints (usually... The Maxwell-Boltzmann distribution is a probability distribution with applications in physics and chemistry. ... For other topics related to Einstein see Einstein (disambig) In statistical mechanics, Bose-Einstein statistics determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium. ... In statistical mechanics, Fermi-Dirac statistics determines the statistical distribution of fermions over the energy states for a system in thermal equilibrium. ... In probability theory and statistics, the geometric distribution is either of two discrete probability distributions: the probability distribution of the number X of Bernoulli trials needed to get one success, supported on the set { 1, 2, 3, ...}, or the probability distribution of the number Y = X âˆ’ 1 of failures before... Download high resolution version (1300x975, 152 KB) Wikipedia does not have an article with this exact name. ... Download high resolution version (1300x975, 152 KB) Wikipedia does not have an article with this exact name. ... In probability theory and statistics, the Poisson distribution is a discrete probability distribution. ... In probability and statistics, the logarithmic distribution (also known as the logarithmic series distribution) is a discrete probability distribution. ... In probability and statistics the negative binomial distribution is a discrete probability distribution. ... In the parabolic fractal distribution, the logarithm of the frequency or size of entities in a population is a quadratic polynomial of the logarithm of the rank. ... In probability theory and statistics, the Poisson distribution is a discrete probability distribution. ... Download high resolution version (1300x975, 225 KB) Some probability mass functions for the Skellam distribution File links The following pages link to this file: Probability distribution Skellam distribution Categories: User-created public domain images ... Download high resolution version (1300x975, 225 KB) Some probability mass functions for the Skellam distribution File links The following pages link to this file: Probability distribution Skellam distribution Categories: User-created public domain images ... The Skellam distribution is the discrete probability distribution of the difference N1 − N2 of two correlated or uncorrelated random variables N1 and N2 having Poisson distributions with different expected values μ1 and μ2. ... The Skellam distribution is the discrete probability distribution of the difference N1 − N2 of two correlated or uncorrelated random variables N1 and N2 having Poisson distributions with different expected values μ1 and μ2. ... In probability and statistics, the Yule-Simon distribution is a discrete probability distribution. ... In probability theory and statistics, the zeta distribution 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. ...

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.

Download high resolution version (1300x975, 214 KB) See the image on the commons for gnuplot source. ... Download high resolution version (1300x975, 214 KB) See the image on the commons for gnuplot source. ... 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 and are parameters that must be greater than zero and is the beta function. ... 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 and are parameters that must be greater than zero and is the beta function. ... Download high resolution version (1300x975, 39 KB) Wikipedia does not have an article with this exact name. ... Download high resolution version (1300x975, 39 KB) Wikipedia does not have an article with this exact name. ... In mathematics, the continuous uniform distributions are probability distributions such that all intervals of the same length are equally probable. ... In mathematics, the continuous uniform distributions are probability distributions such that all intervals of the same length are equally probable. ... In mathematics, the uniform distributions are simple probability distributions. ... The Dirac delta function, sometimes referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere. ... In mathematics, a degenerate distribution is the probability distribution of a random variable which always has the same value. ... In probability theory and statistics, Kumaraswamys double bounded distribution is as versatile as the Beta distribution, but much simpler to use especially in simulation studies as it has a simple closed form solution for both its pdf and cdf. ... In probability theory and statistics, the triangular distribution is a continuous probability distribution. ... In probability theory and statistics, the von Mises distribution is a continuous probability distribution. ... The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution supported on the interval [−R, R] the graph of whose probability density function f is a semicircle of radius R centered at (0, 0) and then suitably normalized (so that it is really a semi-ellipse... In probability theory and statistics, a random matrix is a matrix-valued random variable. ...

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.
    • The noncentral F-distribution
  • The Gamma distribution, which describes the time until n consecutive rare random events occur in a process with no memory.
  • Fisher's z-distribution
  • The half-normal 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.

Download high resolution version (1300x975, 147 KB) Wikipedia does not have an article with this exact name. ... Download high resolution version (1300x975, 147 KB) Wikipedia does not have an article with this exact name. ... In probability theory and statistics, the chi-square distribution (also chi-squared distribution), or χ2  distribution, is one of the theoretical probability distributions most widely used in inferential statistics, i. ... In probability theory and statistics, the chi distribution is a continuous probability distribution. ... In probability theory and statistics, the noncentral chi distribution is a generalization of the chi distribution. ... In probability theory and statistics, the chi-square distribution (also chi-squared distribution), or χ2  distribution, is one of the theoretical probability distributions most widely used in inferential statistics, i. ... Use the Chi Square distribution to establish whether sample data could be drawn from a parent population of a certain, model distribution. ... Statistics is a broad mathematical discipline which studies ways to collect, summarize and draw conclusions from data. ... In probability and statistics, the inverse-chi-square distribution is the probability distribution of a random variable whose inverse has a chi-square distribution. ... In probability theory and statistics, the noncentral chi-square-distribution is a continuous probability distribution that is a generalization of the chi-square distribution. ... The scale-inverse-chi-square distribution arises in Bayesian statistics (spam filtering in particular). ... Download high resolution version (1300x975, 118 KB) See the image on the commons for gnuplot source. ... Download high resolution version (1300x975, 118 KB) See the image on the commons for gnuplot source. ... In probability theory and statistics, the exponential distributions are a class of continuous probability distribution. ... In probability theory and statistics, the exponential distributions are a class of continuous probability distribution. ... In statistics and probability, the F-distribution is a continuous probability distribution. ... In statistics, analysis of variance (ANOVA) is a collection of statistical models and their associated procedures which compare means by splitting the overall observed variance into different parts. ... In probability theory and statistics, the noncentral F-distribution is a continuous probability distribution that is a generalization of the (ordinary) F-distribution. ... Download high resolution version (1300x975, 158 KB) See the image on the commons for gnuplot source. ... Download high resolution version (1300x975, 158 KB) See the image on the commons for gnuplot source. ... In probability theory and statistics, the gamma distribution is a continuous probability distribution. ... In probability theory and statistics, the gamma distribution is a continuous probability distribution. ... The Erlang distribution is a continuous probability distribution with wide applicability primarily due to its relation to the exponential and Gamma distributions. ... The inverse gamma distribution has the probability density function over the support with shape parameter and scale parameter . ... In probability theory and statistics, the Lévy distribution, named after Paul Pierre Lévy, is one of the few distributions that are stable and that have probability density functions that are analytically expressible. ... In probability and statistics, the log-normal distribution is the probability distribution of any random variable whose logarithm is normally distributed (the base of the logarithmic function is immaterial in that loga X is normally distributed if and only if logb X is normally distributed). ... Download high resolution version (1300x975, 119 KB) Wikipedia does not have an article with this exact name. ... Download high resolution version (1300x975, 119 KB) Wikipedia does not have an article with this exact name. ... The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law probability distribution found in a large number of real-world situations. ... The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law probability distribution found in a large number of real-world situations. ... In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution. ... In probability theory and statistics, the Rice distribution distribution is a continuous probability distribution. ... In probability theory, the Type-2 Gumbel distribution function is for . Based on gsl-ref_19. ... In probability theory and statistics, the Weibull distribution (named after Waloddi Weibull) is a continuous probability distribution with the probability density function where and is the shape parameter and is the scale parameter of the distribution. ...

Supported on the whole real line

Download high resolution version (1300x975, 154 KB) Wikipedia does not have an article with this exact name. ... Download high resolution version (1300x975, 154 KB) Wikipedia does not have an article with this exact name. ... The Cauchy-Lorentz distribution, named after Augustin Cauchy, is a continuous probability distribution with probability density function where x0 is the location parameter, specifying the location of the peak of the distribution, and γ is the scale parameter which specifies the half-width at half-maximum (HWHM). ... Download high resolution version (1300x975, 135 KB) Wikipedia does not have an article with this exact name. ... Download high resolution version (1300x975, 135 KB) Wikipedia does not have an article with this exact name. ... In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. ... Download high resolution version (1300x975, 186 KB) Plot of some symmetric centered Levy distributions File links The following pages link to this file: Probability distribution Categories: User-created public domain images | Probability distributions images ... Download high resolution version (1300x975, 186 KB) Plot of some symmetric centered Levy distributions File links The following pages link to this file: Probability distribution Categories: User-created public domain images | Probability distributions images ... A set of four symmetric centered Lévy distributions with scale factor c=1. ... Download high resolution version (1300x975, 135 KB) Wikipedia does not have an article with this exact name. ... Download high resolution version (1300x975, 135 KB) Wikipedia does not have an article with this exact name. ... The normal distribution, also called Gaussian distribution, is an extremely important probability distribution in many fields. ... A Beta Prime Distribution is a distribution with probability function: where is a Beta function. ... The Cauchy-Lorentz distribution, named after Augustin Cauchy, is a continuous probability distribution with probability density function where x0 is the location parameter, specifying the location of the peak of the distribution, and γ is the scale parameter which specifies the half-width at half-maximum (HWHM). ... In probability theory (and especially gambling), the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical... In probability theory and statistics, the variance of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically are. ... ... In physics, resonance is the tendency of a system to absorb more oscillatory energy when the frequency of the oscillations matches the systems natural frequency of vibration (its resonant frequency) than it does at other frequencies. ... A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from an excess or deficiency of photons in a narrow frequency range, compared with the nearby frequencies. ... The Stark effect is the splitting of a spectral line into several components in the presence of an electric field. ... In probability theory and statistics the Gumbel distribution is used to find the minimum (or the maximum) of a number of samples of various distributions. ... This article needs cleanup. ... In probability theory and statistics, the generalized extreme value distribution (GEV) is a family of continuous probability distributions developed within extreme value theory to combine the Gumbel, Fréchet and Weibull families also known as type I, II and III extreme value distributions. ... In probability theory and statistics, the hyperbolic secant distribution is a continuous probability distribution whose probability density function and characteristic function are proportional to the hyperbolic secant function. ... The probability distribution for Landau random variates is defined analytically by the complex integral, For numerical purposes it is more convenient to use the following equivalent form of the integral, From GSL manual, used under GFDL. ... In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. ... In probability theory, a Lévy skew alpha-stable distribution or just stable distribution, developed by Paul Lévy, is a probability distribution where sums of independent identically distributed random variables have the same distribution as the original. ... The normal distribution, also called Gaussian distribution, is an extremely important probability distribution in many fields. ... Central limit theorems are a set of weak-convergence results in probability theory. ... In probability and statistics, the t-distribution or Students t-distribution is a probability distribution that arises in the problem of estimating the mean of a normally distributed population when the sample size is small. ... In probability theory, the Type-1 Gumbel distribution function is for . Reference Taken from the gsl-ref_19. ... In spectroscopy, the Voigt profile is a spectral line profile named after Woldemar Voigt and found in all branches of spectroscopy in which a spectral line is broadened by two types of mechanisms, one of which alone would produce a Doppler profile, and the other of which would produce a... The normal distribution, also called Gaussian distribution, is an extremely important probability distribution in many fields. ... The Cauchy-Lorentz distribution, named after Augustin Cauchy, is a continuous probability distribution with probability density function where x0 is the location parameter, specifying the location of the peak of the distribution, and γ is the scale parameter which specifies the half-width at half-maximum (HWHM). ... A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from an excess or deficiency of photons in a narrow frequency range, compared with the nearby frequencies. ... The Doppler profile is a spectral line profile which results from the thermal motion of the emitting atom or molecule. ...

Joint distributions

For any set of independent random variables the probability density function of the joint distribution is the product of the individual ones. In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ...


Two or more random variables on the same sample space

In probability and statistics, the Dirichlet distribution (after Johann Peter Gustav Lejeune Dirichlet) is a continuous multivariate probability distribution. ... 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 and are parameters that must be greater than zero and 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 five evolutionary forces: natural selection, genetic drift, mutation, migration and nonrandom mating. ... In probability theory, the multinomial distribution is a generalization of the binomial distribution. ... See binomial (disambiguation) for a list of other topics using that name. ... In probability theory and statistics, a multivariate normal distribution, also sometimes called a multivariate Gaussian distribution (in honor of Carl Friedrich Gauss, who was not the first to write about the normal distribution) is a specific probability distribution. ... The normal distribution, also called Gaussian distribution, is an extremely important probability distribution 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. ...

See also

In statistics, a copula is a probability distribution on a unit cube [0, 1]n for which every marginal distribution is uniform on the interval [0, 1]. Sklars theorem is as follows. ... 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 variable X takes on a value less than or... In statistics, a likelihood function is a conditional probability function considered a function of its second argument with its first argument held fixed, thus: and also any other function proportional to such a function. ... Please add any Wikipedia articles related to statistics that are not already on this list. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ... A random variable can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result. ... In statistics, a histogram is a graphical display of tabulated frequencies. ...


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