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.

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 economics, the Lorenz curve is a graphical representation of the cumulative distribution function of the empirical probability distribution of wealth ; it is a graph showing the proportion of the distribution assumed by the bottom y % of the values.

For a probability density function f ( x ) with the cumulative distribution function F ( x ), the Lorenz curve L ( F ( x )) is given by:

If the random variable is real-valued ( or more generally, if a total order is defined for its possible values ), the cumulative distribution function gives the probability that the random variable is no larger than a given value ; in the real-valued case it is the integral of the density.

Both concepts can be united using a cumulative distribution function ( CDF ), which describes the probability that an outcome will be less than or equal to a specified value.

For n independent and identically distributed continuous random variables X < sub > 1 </ sub >, X < sub > 2 </ sub >, ..., X < sub > n </ sub > with cumulative distribution function G ( x ) and probability density function g ( x ) the range of the X < sub > i </ sub > is the range of a sample of size n from a population with distribution function G ( x ).

For n nonidentically distributed independent continuous random variables X < sub > 1 </ sub >, X < sub > 2 </ sub >, ..., X < sub > n </ sub > with cumulative distribution functions G < sub > 1 </ sub >( x ), G < sub > 2 </ sub >( x ), ..., G < sub > n </ sub >( x ) and probability density functions g < sub > 1 </ sub >( x ), g < sub > 2 </ sub >( x ), ..., g < sub > n </ sub >( x ), the range has cumulative distribution function

For n independent and identically distributed discrete random variables X < sub > 1 </ sub >, X < sub > 2 </ sub >, ..., X < sub > n </ sub > with cumulative distribution function G ( x ) and probability mass function g ( x ) the range of the X < sub > i </ sub > is the range of a sample of size n from a population with distribution function G ( x ).

As a probability measure on R, the delta measure is characterized by its cumulative distribution function, which is the unit step function

The cumulative probability of finishing a game of Chutes and Ladders by turn N

) is a basic method for pseudo-random number sampling, i. e. for generating sample numbers at random from any probability distribution given its cumulative distribution function ( cdf ).

The cumulative distribution function ( cdf ) F ( x < sub > 0 </ sub >) of a random vector x is defined as the probability that all components of x are less than or equal to the corresponding values in the vector x < sub > 0 </ sub >.

1 for all x ≥ b, then the function can be taken to represent a cumulative distribution function for a random variable which is neither a discrete random variable ( since the probability is zero for each point ) nor an absolutely continuous random variable ( since the probability density is zero everywhere it exists ).

Note that the cumulative column contains the probability of being dealt that hand or any of the hands ranked higher than it.

The integral of any smooth, positive, " bump-shaped " function will be sigmoidal, thus the cumulative distribution functions for many common probability distributions are sigmoidal.

When using probability theory to analyze order statistics of random samples from a continuous distribution, the cumulative distribution function is used to reduce the analysis to the case of order statistics of the uniform distribution.

The logit and probit are both sigmoid functions with a domain between 0 and 1, which makes then both quantile functions — i. e. inverses of the cumulative distribution function ( CDF ) of a probability distribution.

Thus, it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions.

A revised version, called cumulative prospect theory overcame this problem by using a probability weighting function derived from Rank-dependent expected utility theory.

One can also express it as an implied percentage probability, via, where is the standard normal cumulative distribution function ; thus a moneyness of 0 yields a 50 % probability of expiring ITM, while a moneyness of 1 yields an approximately 84 % probability of expiring ITM.