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.

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 the internal conversion process, the wavefunction of an inner shell electron penetrates the nucleus ( i. e. there is a finite probability of the electron in an s atomic orbital being found in the nucleus ) and when this occurs, the electron may couple to the excited state of the nucleus and take the energy of the nuclear transition directly, without an intermediate gamma ray being first produced.

If the distribution of X is either an exponential distribution or a gamma distribution, then the conditional distributions of Y | N are gamma distributions in which the shape parameters are proportional to N. This shows that the formulation of the " compound Poisson distribution " outlined above is essentially the same as the more general class of compound probability distributions.

At each interface between materials, the probability of transition radiation increases with the relativistic gamma factor.

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to the gamma distribution.

The inverse gamma distribution's probability density function is defined over the support

The probability that an emitted gamma ray will interact with the detector and produce a count is the efficiency of the detector.

Absolute efficiency values represent the probability that a gamma ray of a specified energy passing through the detector will interact and be detected.

Suppose X < sub > 1 </ sub >, ..., X < sub > n </ sub > are independent identically distributed random variables with a gamma distribution with probability density function

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

We devote a chapter to the binomial distribution not only because it is a mathematical model for an enormous variety of real life phenomena, but also because it has important properties that recur in many other probability models.

When each number of successes X is paired with its probability of occurrence Af, the set of pairs Af, is a probability function called a binomial distribution.

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.

This is specified by a distribution function Af such that the probability that Af lies in some region D of the stage space is Af.

The analysis of variance can be presented in terms of a linear model, which makes the following assumptions about the probability distribution of the responses:

Max Born suggested that the electron's position needed to be described by a probability distribution which was connected with finding the electron at some point in the wave-function which described its associated wave packet.

The general consensus in the scientific community, however, was to associate this type of complexity with Kolmogorov, who was concerned with randomness of a sequence, while Algorithmic Probability became associated with Solomonoff, who focused on prediction using his invention of the universal a priori probability distribution.

In chemistry, physics, and mathematics, the Boltzmann distribution ( also called the Gibbs Distribution ) is a certain distribution function or probability measure for the distribution of the states of a system.

If there are g ( E ) dE states with energy E to E + dE, then the Boltzmann distribution predicts a probability distribution for the energy:

Ramsey and Savage noted that the individual agent's probability distribution could be objectively studied in experiments.

# REDIRECT Probability distribution # Continuous probability distribution

# REDIRECTProbability distribution # Continuous probability distribution