Sample areas in the new investigations were selected strictly by application of the principles of probability theory, so as to be representative of the total population of defined areas within calculable limits.

This list could be expanded to include most fields of mathematics, including measure theory, ergodic theory, probability, representation theory, and differential geometry.

Occasionally, " almost all " is used in the sense of " almost everywhere " in measure theory, or in the closely related sense of " almost surely " in probability theory.

The concept and theory of Kolmogorov Complexity is based on a crucial theorem first discovered by Ray Solomonoff, who published it in 1960, describing it in " A Preliminary Report on a General Theory of Inductive Inference " as part of his invention of algorithmic probability.

In information theory, one bit is typically defined as the uncertainty of a binary random variable that is 0 or 1 with equal probability, or the information that is gained when the value of such a variable becomes known.

Pascal was an important mathematician, helping create two major new areas of research: he wrote a significant treatise on the subject of projective geometry at the age of sixteen, and later corresponded with Pierre de Fermat on probability theory, strongly influencing the development of modern economics and social science.

In computational complexity theory, BPP, which stands for bounded-error probabilistic polynomial time is the class of decision problems solvable by a probabilistic Turing machine in polynomial time, with an error probability of at most 1 / 3 for all instances.

In computational complexity theory, BQP ( bounded error quantum polynomial time ) is the class of decision problems solvable by a quantum computer in polynomial time, with an error probability of at most 1 / 3 for all instances.

Following the work on expected utility theory of Ramsey and von Neumann, decision-theorists have accounted for rational behavior using a probability distribution for the agent.

Johann Pfanzagl completed the Theory of Games and Economic Behavior by providing an axiomatization of subjective probability and utility, a task left uncompleted by von Neumann and Oskar Morgenstern: their original theory supposed that all the agents had the same probability distribution, as a convenience.

The " Ramsey test " for evaluating probability distributions is implementable in theory, and has kept experimental psychologists occupied for a half century.

Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, and combinatorics also has many applications in optimization, computer science, ergodic theory and statistical physics.

In part, the growth was spurred by new connections and applications to other fields, ranging from algebra to probability, from functional analysis to number theory, etc.

Analytic combinatorics concerns the enumeration of combinatorial structures using tools from complex analysis and probability theory.

In probability theory and statistics, the cumulative distribution function ( CDF ), or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x.

This is totally spurious, since no matter who measured first the other will measure the opposite spin despite the fact that ( in theory ) the other has a 50 % ' probability ' ( 50: 50 chance ) of measuring the same spin, unless data about the first spin measurement has somehow passed faster than light ( of course TI gets around the light speed limit by having information travel backwards in time instead ).

In the computer science subfield of algorithmic information theory, a Chaitin constant ( Chaitin omega number ) or halting probability is a real number that informally represents the probability that a randomly constructed program will halt.

Archaeoastronomy uses a variety of methods to uncover evidence of past practices including archaeology, anthropology, astronomy, statistics and probability, and history.

covers statistical study, descriptive statistics ( collection, description, analysis, and summary of data ), probability, and the binomial and normal distributions, test of hypotheses and confidence intervals, linear regression, and correlation.

In Bayesian statistics, a probability can be assigned to a hypothesis that can differ from 0 or 1 if the truth value is uncertain.

For objectivists, probability objectively measures the plausibility of propositions, i. e. the probability of a proposition corresponds to a reasonable belief everyone ( even a " robot ") sharing the same knowledge should share in accordance with the rules of Bayesian statistics, which can be justified by requirements of rationality and consistency.

After the 1920s, " inverse probability " was largely supplanted by a collection of methods that came to be called frequentist statistics.

* Conjugate prior, in Bayesian statistics, a family of probability distributions that contains a prior and the posterior distributions for a particular likelihood function ( particularly for one-parameter exponential families )

It has applications that include probability, statistics, computer vision, image and signal processing, electrical engineering, and differential equations.

This generally means that descriptive statistics, unlike inferential statistics, are not developed on the basis of probability theory.

As with other branches of statistics, experimental design is pursued using both frequentist and Bayesian approaches: In evaluating statistical procedures like experimental designs, frequentist statistics studies the sampling distribution while Bayesian statistics updates a probability distribution on the parameter space.

Fourier analysis has many scientific applications – in physics, partial differential equations, number theory, combinatorics, signal processing, imaging, probability theory, statistics, option pricing, cryptography, numerical analysis, acoustics, oceanography, sonar, optics, diffraction, geometry, protein structure analysis and other areas.

* In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions.

The gamma function is a component in various probability-distribution functions, and as such it is applicable in the fields of probability and statistics, as well as combinatorics.

* fundamental applications of probability and statistics

Information theory is based on probability theory and statistics.

The most complicated aspect of the insurance business is the actuarial science of ratemaking ( price-setting ) of policies, which uses statistics and probability to approximate the rate of future claims based on a given risk.

In statistics, the Kolmogorov – Smirnov test ( K – S test ) is a nonparametric test for the equality of continuous, one-dimensional probability distributions that can be used to compare a sample with a reference probability distribution ( one-sample K – S test ), or to compare two samples ( two-sample K – S test ).

** Normal dynamics, is a stochastic motion having a Gaussian probability density function in position with variance MSD that follows, MSD ~ t, where MSD is the mean squared displacement of the process, and t is the time the process is seen ( normal dynamics and Brownian dynamics are very similar ; the term used depends on the field )

will give the probability that a single sample taken from a random process with zero-mean and unit-variance Gaussian probability density function will be greater or equal to.

For sufficiently nice prior probabilities, the Bernstein-von Mises theorem gives that in the limit of infinite trials and the posterior converges to a Gaussian distribution independent of the initial prior under some conditions firstly outlined and rigorously proven by Joseph Leo Doob in 1948, namely if the random variable in consideration has a finite probability space.

In probability theory and statistics, the multivariate normal distribution or multivariate Gaussian distribution, is a generalization of the one-dimensional ( univariate ) normal distribution to higher dimensions.

A commonly used symmetric jumping distribution is a Gaussian distribution centered at, which tends to move to points nearby ( and thus explores the probability space using a random walk ).

* Normal distribution, the Gaussian continuous probability distribution

The force η ( t ) has a Gaussian probability distribution with correlation function

* The Gauss – Markov theorem in mathematical statistics ( In this theorem, one does not assume the probability distributions are Gaussian.

Additionally, the amplitude of the signal has very nearly a Gaussian probability density function.

In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence that arise in probability, such as the Edgeworth series ; in combinatorics, as an example of an Appell sequence, obeying the umbral calculus ; in numerical analysis as Gaussian quadrature ; in finite element methods as Shape Functions for beams ; and in physics, where they give rise to the eigenstates of the quantum harmonic oscillator.

* Gaussian process, a stochastic process that is associated with the Gaussian probability distribution

This integral is 1 if and only if a = 1 /( c √( 2π )), and in this case the Gaussian is the probability density function of a normally distributed random variable with expected value μ = b and variance σ < sup > 2 </ sup > = c < sup > 2 </ sup >.

A Gaussian process can be used as a prior probability distribution over functions in Bayesian inference.

The end-to-end distance probability distribution function of a Gaussian chain is non-zero only for r > 0.

Position space probability density of an initially Gaussian state moving in one dimension at minimally uncertain, constant momentum in free space.

Often, in popular culture, an endangering huge wave is loosely denoted as a rogue wave, while it has not been ( and most often cannot be ) established that the reported event is a rogue wave in the scientific sense — i. e. of a very different nature in characteristics as the surrounding waves in that sea state and with very low probability of occurrence ( according to a Gaussian process description as valid for linear wave theory ).

The probability density function of having an error of a given size can be modelled by a Gaussian function ; the mean value will be the relative sent value, and its variance will be given by:

The significance of the central pixel may be increased, as it approximates the properties of noise with a Gaussian probability distribution:

The probability of a given projection is, as before, given by the product of the likelihood of the data under the Gaussian noise model with the prior on the deformation parameter.

* Generalized inverse Gaussian distribution, a distribution in probability theory

In statistics, a probability distribution is said to have a long tail if a larger share of population rests within its tail than would under a " normal " or Gaussian distribution.

Gaussian noise is statistical noise that has its probability density function equal to that of the normal distribution, which is also known as the Gaussian distribution.