Probability spaceIn probability theory, a probability space or a probability triple is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models the throwing of a die. A probability space consists of three elements: A sample space, , which is the set of all possible outcomes. An event space, which is a set of events, , an event being a set of outcomes in the sample space. A probability function, , which assigns each event in the event space a probability, which is a number between 0 and 1.
Interaction designInteraction design, often abbreviated as IxD, is "the practice of designing interactive digital products, environments, systems, and services." While interaction design has an interest in form (similar to other design fields), its main area of focus rests on behavior. Rather than analyzing how things are, interaction design synthesizes and imagines things as they could be. This element of interaction design is what characterizes IxD as a design field, as opposed to a science or engineering field.
Chi-squared distributionIn probability theory and statistics, the chi-squared distribution (also chi-square or -distribution) with degrees of freedom is the distribution of a sum of the squares of independent standard normal random variables. The chi-squared distribution is a special case of the gamma distribution and is one of the most widely used probability distributions in inferential statistics, notably in hypothesis testing and in construction of confidence intervals.
Fixed point (mathematics)hatnote|1=Fixed points in mathematics are not to be confused with other uses of "fixed point", or stationary points where math|1=f(x) = 0. In mathematics, a fixed point (sometimes shortened to fixpoint), also known as an invariant point, is a value that does not change under a given transformation. Specifically for functions, a fixed point is an element that is mapped to itself by the function. Formally, c is a fixed point of a function f if c belongs to both the domain and the codomain of f, and f(c) = c.
Probability axiomsThe Kolmogorov axioms are the foundations of probability theory introduced by Russian mathematician Andrey Kolmogorov in 1933. These axioms remain central and have direct contributions to mathematics, the physical sciences, and real-world probability cases. An alternative approach to formalising probability, favoured by some Bayesians, is given by Cox's theorem. The assumptions as to setting up the axioms can be summarised as follows: Let be a measure space with being the probability of some event , and .
Fixed-point theoremIn mathematics, a fixed-point theorem is a result saying that a function F will have at least one fixed point (a point x for which F(x) = x), under some conditions on F that can be stated in general terms. The Banach fixed-point theorem (1922) gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point.
Bayesian probabilityBayesian probability (ˈbeɪziən or ˈbeɪʒən ) is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation representing a state of knowledge or as quantification of a personal belief. The Bayesian interpretation of probability can be seen as an extension of propositional logic that enables reasoning with hypotheses; that is, with propositions whose truth or falsity is unknown.
Poisson distributionIn probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. It is named after French mathematician Siméon Denis Poisson ('pwɑːsɒn; pwasɔ̃). The Poisson distribution can also be used for the number of events in other specified interval types such as distance, area, or volume.
Brouwer fixed-point theoremBrouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function mapping a nonempty compact convex set to itself there is a point such that . The simplest forms of Brouwer's theorem are for continuous functions from a closed interval in the real numbers to itself or from a closed disk to itself. A more general form than the latter is for continuous functions from a nonempty convex compact subset of Euclidean space to itself.
Expected valueIn probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a large number of independently selected outcomes of a random variable. The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by integration.
