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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. ...


The concept of the probability distribution and the random variables which they describe underlies the mathematical discipline of probability theory, and the science of statistics. There is spread or variability in almost any value that can be measured in a population (e.g. height of people, durability of a metal, etc.); almost all measurements are made with some intrinsic error; in physics many processes are described probabilistically, from the kinetic properties of gases to the quantum mechanical description of fundamental particles. For these and many other reasons, simple numbers are often inadequate for describing a quantity, while probability distributions are often more appropriate models. There are, however, considerable mathematical complications in manipulating probability distributions, since most standard arithmetic and algebraic manipulations cannot be applied. 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. ... Probability theory is the branch of mathematics concerned with analysis of random phenomena. ... This article is about the field of statistics. ... Measurement is the determination of the size or magnitude of something. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... Kinetic theory or kinetic theory of gases attempts to explain macroscopic properties of gases, such as pressure, temperature, or volume, by considering their molecular composition and motion. ... Fig. ... ... Look up number in Wiktionary, the free dictionary. ... Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. ... Algebraic chess notation is the method used today by all competition chess organizations and most books, magazines, and newspapers to record and describe the play of chess games. ...

Contents

Rigorous definitions

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. ... “Superset” 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, Ω 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 ε, no matter how small, there is a positive number δ 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 f′(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.

In physics, the Boltzmann distribution predicts the distribution function for the fractional number of particles Ni / N occupying a set of states i which each has energy Ei: where is the Boltzmann constant, T is temperature (assumed to be a sharply well-defined quantity), is the degeneracy, or number of... 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 introduction to this article provides insufficient context for those unfamiliar with the subject matter. ... 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 that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate, and are independent of the time since the last event. ... 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 that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate, and are independent of the time since the last event. ... 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 α and β are parameters that must be greater than zero and B 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 α and β are parameters that must be greater than zero and B 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 probability theory and statistics, the continuous uniform distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distributions support are equally probable. ... In probability theory and statistics, the continuous uniform distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distributions support are equally probable. ... In mathematics, the uniform distributions are simple probability distributions. ... The Dirac delta or Diracs delta, often 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 and the value zero elsewhere. ... In mathematics, a degenerate distribution is the probability distribution of a random variable which always has the same value. ... The 5-parameter Fisher-Bingham distribution or Kent distribution is a probability distribution on the three-dimensional sphere. ... 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 with lower limit a, mode c and upper limit b. ... In probability and statistics, the truncated normal distribution is the probability distribution of a normally distributed random variable whose value is either bounded below or above (or both). ... In probability theory and statistics, the U-quadratic distribution is a continuous probability distribution defined by a unique quadratic function with lower limit a and upper limit b. ... In probability theory and statistics, the von Mises distribution is a continuous probability distribution. ... Points sampled from three von Mises-Fisher distributions on the sphere (blue: , green: , red: ). The mean directions are shown with arrows. ... 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. (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.

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. ... This article is about the mathematics of the chi-square distribution. ... 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. ... This article is about the mathematics of the chi-square distribution. ... The Goodness of fit of a statistical model describes how well it fits a set of observations. ... This article is about the field of statistics. ... 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 or noncentral distribution 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. ... A Beta Prime Distribution is a distribution with probability function: where is a Beta function. ... 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 two-parameter family of continuous probability distributions. ... In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. ... The Erlang distribution is a continuous probability distribution with wide applicability primarily due to its relation to the exponential and Gamma distributions. ... Queueing theory (also commonly spelled queuing theory) is the mathematical study of waiting lines (or queues). ... The inverse gamma distribution has the probability density function over the support with shape parameter and scale parameter . ... The folded normal distribution is a probability distribution related to the normal distribution. ... The probability density function of the inverse Gaussian distribution is given by The Wald distribution is simply another name for the inverse Gaussian distribution. ... 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 single-tailed probability distribution of any random variable whose logarithm 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. ... This article or section is incomplete and may require expansion and/or cleanup. ... 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. ... The particle size distribution[1] (PSD) of a powder or granular material is a list of values or a mathematical function that defines the relative amounts of particles present, sorted according to size. ... The word grinding can mean many things: Grinding is a manufacturing process that uses friction with a rough surface to wear away or smooth the surface of a work piece - see grinding machine. ... Milling may refer to: Grinding grain and other materials in a mill Cutting and shaping materials into products with milling machines Miller Category: ... Death by crushing or pressing is a method of execution which has a long history during which the techniques used varied greatly from place to place. ...

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 the Gaussian distribution, is an important family of continuous probability distributions, applicable 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). ... In probability theory 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 as the outcome of the random trial when identical odds are... This article is about mathematics. ... ... This article is about resonance in physics. ... 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 (named after Emil Julius Gumbel (1891–1966)) is used to find the minimum (or the maximum) of a number of samples of various distributions. ... This article needs cleanup. ... Fishers z-distribution is the distribution of half the logarithm of a F distribution variate: It was first described by Ronald Fisher in a paper delivered at the International Mathematical Congress of 1924 in Toronto, entitled On a distribution yielding the error functions of several well-known statistics. Nowadays... 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. ... The hyperbolic distribution is a continuous probability distribution that is characterized by the fact that the logarithm of the probability density function is a hyperbola. ... 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 the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... A central limit theorem is any of a set of weak-convergence results in probability theory. ... This article or section is incomplete and may require expansion and/or cleanup. ... 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. ... Where the hell is this ? ... 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 the Gaussian distribution, is an important family of continuous probability distributions, applicable 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 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 α. Clockwise from top left: α=(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 α and β 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. ...

See also

In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ... In statistics, a copula is a multivariate cumulative distribution function defined on the n-dimensional unit cube [0, 1]n such that 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 random variable X takes on a value less than... Look up likelihood in Wiktionary, the free dictionary. ... Please add any Wikipedia articles related to statistics that are not already on this list. ... In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ... For the histogram used in digital image processing, see Color histogram. ... Inverse transform sampling , also known as the probability integral transform, is a method of sampling a number at random from any probability distribution given its cumulative distribution function (cdf). ... In mathematics, the Riemann-Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. ... A probability distribution describes the values and probabilities that a random event can take place. ... A logarithmic scale bar. ... 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 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. ... Often confused with the multinomial distribution. ... // 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. ... In probability theory and statistics, the Rademacher distribution is a discrete probability distribution. ... 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. ... 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. ... A probability distribution describes the values and probabilities that a random event can take place. ... In physics, the Boltzmann distribution predicts the distribution function for the fractional number of particles Ni / N occupying a set of states i which each has energy Ei: where is the Boltzmann constant, T is temperature (assumed to be a sharply well-defined quantity), is the degeneracy, or number of... In probability theory, a compound Poisson distribution is the probability distribution of a Poisson-distibuted number of independent identically-distributed random variables. ... The discrete phase-type distribution is a probability distribution that results from a system of one or more inter-related geometric distributions occurring in sequence, or phases. ... In mathematics, the Gauss-Kuzmin distribution gives the probability distribution of the occurrence of a given integer in the continued fraction expansion of an arbitrary real number. ... 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... 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 that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate, and are independent of the time since the last event. ... 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. ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... 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 α and β are parameters that must be greater than zero and B is the beta function. ... 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 raised cosine distribution is a probability distribution supported on the interval []. The probability density function is for and zero otherwise. ... In probability theory and statistics, the triangular distribution is a continuous probability distribution with lower limit a, mode c and upper limit b. ... In probability theory and statistics, the U-quadratic distribution is a continuous probability distribution defined by a unique quadratic function with lower limit a and upper limit b. ... In probability theory and statistics, the continuous uniform distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distributions support are equally probable. ... 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 mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... A Beta Prime Distribution is a distribution with probability function: where is a Beta function. ... This article is about the mathematics of the chi-square distribution. ... 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. ... The Erlang distribution is a continuous probability distribution with wide applicability primarily due to its relation to the exponential and Gamma distributions. ... 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. ... This article does not cite its references or sources. ... The folded normal distribution is a probability distribution related to the normal distribution. ... In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. ... 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, the Generalized inverse Gaussian distribution (GIG) is a probability distribution with probability density function It is used extensively in geostatistics, statistical linguistics, finance, etc. ... In probability theory and statistics, the half-logistic distribution is a continuous probability distribution—the distribution of the absolute value of a random variable following the logistic 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. ... In probability theory, a hyper-exponential distribution is a continuous distribution such that the probability density function of the random variable X is given by: Where is an exponentially distributed random variable with rate parameter , and is the probability that X will take on the form of the exponential distribution... The hypoexponential distribution is a generalization of Erlang distribution in the sense that the n exponential distributions may have different rates. ... In probability and statistics, the inverse-chi-square distribution is the probability distribution of a random variable whose inverse has a chi-square distribution. ... The scale-inverse-chi-square distribution arises in Bayesian statistics (spam filtering in particular). ... The probability density function of the inverse Gaussian distribution is given by The Wald distribution is simply another name for the inverse Gaussian distribution. ... 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 single-tailed probability distribution of any random variable whose logarithm is normally distributed. ... The Maxwell–Boltzmann distribution is a probability distribution with applications in physics and chemistry. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... There are very few or no other articles that link to this one. ... In probability theory and statistics, the noncentral chi-square or noncentral distribution is a generalization of the chi-square distribution. ... 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. ... 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. ... In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution. ... The relativistic Breit–Wigner distribution (after Gregory Breit and Eugene Wigner) is a continuous probability distribution with the following probability density function [1]: It is most often used to model resonances (i. ... In probability theory and statistics, the Rice distribution distribution is a continuous probability distribution. ... The shifted Gompertz distribution is the distribution of the largest order statistic of two independent random variables which are distributed exponential and Gompertz with parameters b and b and respectively. ... In probability and statistics, the truncated normal distribution is the probability distribution of a normally distributed random variable whose value is either bounded below or above (or both). ... 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. ... This article or section is in need of attention from an expert on the subject. ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... 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). ... This article needs cleanup. ... The exponential power distribution, also known as the generalized error distribution, takes a scale parameter a and exponent b. ... Fishers z-distribution is the distribution of half the logarithm of a F distribution variate: It was first described by Ronald Fisher in a paper delivered at the International Mathematical Congress of 1924 in Toronto, entitled On a distribution yielding the error functions of several well-known statistics. Nowadays... In probability theory and statistics the Gumbel distribution (named after Emil Julius Gumbel (1891–1966)) is used to find the minimum (or the maximum) of a number of samples of various distributions. ... The generalised hyperbolic distribution is a continuous probability distribution defined by the probability density function where is the modified Bessel function of the second kind. ... 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. ... In probability theory and statistics, the logistic distribution is a continuous probability distribution. ... The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... In probability theory and statistics, the normal-gamma distribution is a four-parameter family of continuous probability distributions. ... The normal-inverse Gaussian distribution is continuous probability distribution that is defined as the normal variance-mean mixture where the mixing density is the inverse Gaussian distribution. ... 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. ... The variance-gamma distribution is continuous probability distribution that is defined as the normal variance-mean mixture where the mixing density is the gamma distribution. ... 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... A probability distribution describes the values and probabilities that a random event can take place. ... 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 probability theory, the multinomial distribution is a generalization of the binomial distribution. ... The multivariate Polya distribution, also called the Dirichlet compound multinomial distribution, is a compound probability distribution, where a probability vector p is drawn from a Dirichlet distribution and a set of discrete samples x is drawn from the multinomial distribution with probability vector p. ... Several images of the probability density of the Dirichlet distribution when K=3 for various parameter vectors α. Clockwise from top left: α=(6, 2, 2), (3, 7, 5), (6, 2, 6), (2, 3, 4). ... In statistics, the generalized Dirichlet distribution (GD) is a generalization of the Dirichlet distribution with a more general covariance structure and twice the number of parameters. ... 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). ... In statistics, a multivariate Student distribution is a multivariate generalization of the Students t-distribution. ... A probability distribution describes the values and probabilities that a random event can take place. ... In statistics, the Inverse Wishart distribution, also the inverse Wishart distribution and inverted Wishart distribution is a probability density function defined on matrices. ... The matrix normal distribution is a probability distribution that is a generalization of the normal distribution. ... 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. ... Circular or directional statistics is the subdiscipline of statistics that deals with circular or directional data. ... In mathematics, a degenerate distribution is the probability distribution of a random variable which always has the same value. ... In probability, a singular distribution is a probability distribution concentrated on a measure zero set where the probability of each point in that set is zero. ... Circular or directional statistics is the subdiscipline of statistics that deals with circular or directional data. ... The 5-parameter Fisher-Bingham distribution or Kent distribution is a probability distribution on the three-dimensional sphere. ... In probability theory and statistics, the von Mises distribution is a continuous probability distribution. ... In mathematics, a degenerate distribution is the probability distribution of a random variable which always has the same value. ... In mathematics, a degenerate distribution is the probability distribution of a random variable which always has the same value. ... The Dirac delta or Diracs delta, often 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 and the value zero elsewhere. ... In probability, a singular distribution is a probability distribution concentrated on a measure zero set where the probability of each point in that set is zero. ... The Cantor distribution is the probability distribution whose cumulative distribution function is the Cantor function. ... In probability and statistics, an exponential family is any class of probability distributions having a certain form. ... In probability theory, especially as that field is used in statistics, a location-scale family is a set of probability distributions on the real line parametrized by a location parameter μ and a scale parameter σ â‰¥ 0; if X is any random variable whose probability distribution belongs to such a family, then... In statistics and information theory, a maximum entropy probability distribution is a probability distribution whose entropy is larger than (or equal to) that of all other members of a specified class of distributions. ... This article or section is incomplete and may require expansion and/or cleanup. ... This article is about the field of statistics. ... Descriptive statistics are used to describe the basic features of the data in a study. ... This article is about mathematical mean. ... In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ... The geometric mean of a collection of positive data is defined as the nth root of the product of all the members of the data set, where n is the number of members. ... This article is about the statistical concept. ... In statistics, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. ... This article is about mathematics. ... In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. ... It has been suggested that this article or section be merged with inferential statistics. ... One may be faced with the problem of making a definite decision with respect to an uncertain hypothesis which is known only through its observable consequences. ... In statistics, a result is significant if it is unlikely to have occurred by chance, given that a presumed null hypothesis is true. ... The power of a statistical test is the probability that the test will reject a false null hypothesis (that it will not make a Type II error). ... In statistics, a null hypothesis is a hypothesis set up to be nullified or refuted in order to support an alternative hypothesis. ... The alternate hypothesis (or maintained hypothesis) and the null hypothesis are the two rival hypotheses whose likelihoods are compared by a statistical hypothesis test. ... Type I errors (or α error, or false positive) and type II errors (β error, or a false negative) are two terms used to describe statistical errors. ... The Z-test is a statistical test used in inference. ... A t test is any statistical hypothesis test in which the test statistic has a Students t distribution if the null hypothesis is true. ... Maximum likelihood estimation (MLE) is a popular statistical method used to make inferences about parameters of the underlying probability distribution from a given data set. ... Compares the various grading methods in a normal distribution. ... In statistical hypothesis testing, the p-value of a random variable T used as a test statistic is the probability that T will assume a value at least as extreme as the observed value tobserved, given that a null hypothesis being considered is true. ... 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. ... Survival analysis is a branch of statistics which deals with death in biological organisms and failure in mechanical systems. ... The survival function, also known as a survivor function or reliability function, is a property of any random variable that maps a set of events, usually associated with mortality or failure of some system, onto time. ... The Kaplan-Meier estimator (also known as the Product Limit Estimator) estimates the survival function from life-time data. ... The logrank test (sometimes called the Mantel-Haenszel test or the Mantel-Cox test) [1] is a hypothesis test to compare the survival distributions of two samples. ... Failure rate is the frequency with which an engineered system or component fails, expressed for example in failures per hour. ... // Proportional hazards models are a sub-class of survival models in statistics. ... The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate, and are independent of the time since the last event. ... 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 . ... Positive linear correlations between 1000 pairs of numbers. ... In statistics, a spurious relationship (or, sometimes, spurious correlation) is a mathematical relationship in which two occurrences have no logical connection, yet it may be implied that they do, due to a certain third, unseen factor (referred to as a confounding factor or lurking variable). The spurious relationship gives an... In statistics, the Pearson product-moment correlation coefficient (sometimes known as the PMCC) (r) is a measure of the correlation of two variables X and Y measured on the same object or organism, that is, a measure of the tendency of the variables to increase or decrease together. ... In statistics, rank correlation is the study of relationships between different rankings on the same set of items. ... In statistics, Spearmans rank correlation coefficient, named after Charles Spearman and often denoted by the Greek letter (rho) or as , is a non-parametric measure of correlation – that is, it assesses how well an arbitrary monotonic function could describe the relationship between two variables, without making any assumptions about... The Kendall tau rank correlation coefficient (or simply the Kendall tau coefficient, Kendalls Ï„ or Tau test(s)) is used to measure the degree of correspondence between two rankings and assessing the significance of this correspondence. ... In statistics, regression analysis examines the relation of a dependent variable (response variable) to specified independent variables (explanatory variables). ... In statistics, linear regression is a regression method that models the relationship between a dependent variable Y, independent variables Xi, i = 1, ..., p, and a random term ε. The model can be written as Example of linear regression with one dependent and one independent variable. ... dataset with approximating polynomials Nonlinear regression in statistics is the problem of fitting a model to multidimensional x,y data, where f is a nonlinear function of x with parameters θ. In general, there is no algebraic expression for the best-fitting parameters, as there is in linear regression. ... Logistic regression is a statistical regression model for Bernoulli-distributed dependent variables. ...

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  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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