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 ).

In a similar way to the concept of skewness, kurtosis is a descriptor of the shape of a probability distribution and, just as for skewness, there are different ways of quantifying it for a theoretical distribution and corresponding ways of estimating it from a sample from a population.

However, it was not known how to construct probability distributions in which the skewness ( standardized third cumulant ) and kurtosis ( standardized fourth cumulant ) could be adjusted equally freely.

Data quality can be assessed in several ways, using different types of analyses: frequency counts, descriptive statistics ( mean, standard deviation, median ), normality ( skewness, kurtosis, frequency histograms, normal probability plots ), associations ( correlations, scatter plots ).

Section 7 is designed to arrest in its incipiency not only the substantial lessening of competition from the acquisition by one corporation of the whole or any part of the stock of a competing corporation, but also to arrest in their incipiency restraints or monopolies in a relevant market which, as a reasonable probability, appear at the time of suit likely to result from the acquisition by one corporation of all or any part of the stock of any other corporation.

The binomial probability distribution may describe the variation that occurs from one set of trials of such a binomial experiment to another.

More generally, suppose that an experiment consists of a number of independent trials, that each trial results in either a `` success '' or a `` non-success '' ( `` failure '' ), and that the probability of success remains constant from trial to trial.

In this case the stage R operating with conditions Af transforms the state of the stream from Af to Af, but only the probability distribution of Af is known.

Since in practice it is not worth contrasting a zero probability with one that is nearly indistinguishable from zero, he prefers to categorize himself as a " de facto atheist ".

# The electrons are never in a single point location, although the probability of interacting with the electron at a single point can be found from the wave function of the electron.

These are processes which create compound nuclei at low excitation energy (~ 10-20 MeV, hence " cold "), leading to a higher probability of survival from fission.

These are processes which create compound nuclei at high excitation energy (~ 40-50 MeV, hence " hot "), leading to a reduced probability of survival from fission and quasi-fission.

Bayesian probability interprets the concept of probability as " an abstract concept, a quantity that we assign theoretically, for the purpose of representing a state of knowledge, or that we calculate from previously assigned probabilities ," in contrast to interpreting it as a frequency or " propensity " of some phenomenon.

Early Bayesian inference, which used uniform priors following Laplace's principle of insufficient reason, was called " inverse probability " ( because it infers backwards from observations to parameters, or from effects to causes ).

Pollution can occur from a number of sources: garbage and industrial debris ; the transportation of petroleum in tankers, increasing the probability of large oil spills ; small oil spills created by large and small vessels, which flush bilge water into the ocean.

E. T. Jaynes, from a Bayesian point of view, pointed out probability is a measure of a human's information about the physical world.

The earliest radiocarbon determinations obtained from key sites such as Oakbank in Loch Tay or Redcastle, Beauly Firth approach the Late Bronze Age-Early Iron Age transition at their widest interpretation at 2 sigma or 95. 4 % probability, falling after c. 800BC and therefore could only be considered Late Bronze Age by the narrowest of margins.

The appearance of interference built up from individual photons could seemingly be explained by assuming that a single photon has its own associated wavefront that passes through both slits, and that the single photon will show up on the detector screen according to the net probability values resulting from the co-incidence of the two probability waves coming by way of the two slits.

The probability " wave " can be said to " pass through space " because the probability values that one can compute from its mathematical representation are dependent on time.