Information: - Probability is the measure of the likelihood that an event will occur. Probability is quantified as a number between 0 and 1 (where 0 indicates impossibility and 1 indicates certainty). The higher the probability of an event, the more certain that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is unbiased, the two outcomes ("head" and "tail") are both equally probable; the probability of "head" equals the probability of "tail". Since no other outcomes are possible, the probability is 1/2 (or 50%), of either "head" or "tail". In other words, the probability of "head" is 1 out of 2 outcomes and the probability of "tail" is also 1 out of 2 outcomes, expressed as 0.5 when converted to decimal, with the above-mentioned quantification system. This type of probability is also called a priori probability. - In mathematics, the Bernoulli scheme or Bernoulli shift is a generalization of the Bernoulli process to more than two possible outcomes. Bernoulli schemes are important in the study of dynamical systems, as most such systems (such as Axiom A systems) exhibit a repellor that is the product of the Cantor set and a smooth manifold, and the dynamics on the Cantor set are isomorphic to that of the Bernoulli shift. This is essentially the Markov partition. The term "shift" is in reference to the shift operator, which may be used to study Bernoulli schemes. The Ornstein isomorphism theorem shows that Bernoulli shifts are isomorphic when their entropy is equal. - In mathematics, and in particular functional analysis, the shift operator or translation operator is an operator that takes a function to its translation . In time series analysis, the shift operator is called the lag operator. - In mathematics, the Ornstein isomorphism theorem is a deep result for ergodic theory. It states that if two different Bernoulli schemes have the same Kolmogorov entropy, then they are isomorphic. The result, given by Donald Ornstein in 1970, is important because it states that many systems previously believed to be unrelated are in fact isomorphic; these include all finite stationary stochastic processes, subshifts of finite type and Markov shifts, Anosov flows and Sinai's billiards, ergodic automorphisms of the "n"-torus, and the continued fraction transform. - In probability and statistics , a Bernoulli process is a finite or infinite sequence of binary random variables , so it is a discrete - time stochastic process that takes only two values , canonically 0 and 1 . The component Bernoulli variables Xi are identical and independent . Prosaically , a Bernoulli process is a repeated coin flipping , possibly with an unfair coin ( but with consistent unfairness ) . Every variable Xi in the sequence is associated with a Bernoulli trial or experiment . They all have the same Bernoulli distribution . Much of what can be said about the Bernoulli process can also be generalized to more than two outcomes ( such as the process for a six - sided dice ) ; this generalization is known as the Bernoulli scheme . The problem of determining the process , given only a limited sample of the Bernoulli trials , may be called the problem of checking whether a coin is fair . - In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a lake. - Coin flipping, coin tossing, or heads or tails is the practice of throwing a coin in the air to choose between two alternatives, sometimes to resolve a dispute between two parties. It is a form of sortition which inherently has only two possible and equally likely outcomes. - In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is conducted. It is named after Jacob Bernoulli, a 17th century Swiss mathematician. - In probability theory and statistics, a probability distribution is a mathematical function that, stated in simple terms, can be thought of as providing the probability of occurrence of different possible outcomes in an experiment. For instance, if the random variable X is used to denote the outcome of a coin toss ('the experiment'), then the probability distribution of X would take the value 0.5 for formula_1, and 0.5 for formula_2. - Analysis is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (384322 B.C.), though "analysis" as a formal concept is a relatively recent development. - Debt, AIDS, Trade, Africa (DATA) was a multinational non-government organization founded in January 2002 in London by U2's Bono along with Bobby Shriver and activists from the Jubilee 2000 Drop the Debt campaign. - Jacob Bernoulli (also known as James or Jacques; 16 August 1705) was one of the many prominent mathematicians in the Bernoulli family. He was an early proponent of Leibnizian calculus and had sided with Leibniz during the LeibnizNewton calculus controversy. He is known for his numerous contributions to calculus, and along with his brother Johann, was one of the founders of the calculus of variations. However, his most important contribution was in the field of probability, where he derived the first version of the law of large numbers in his work "Ars Conjectandi". - Mathematics (from Greek "mathma", knowledge, study, learning) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics. - A Markov partition is a tool used in dynamical systems theory, allowing the methods of symbolic dynamics to be applied to the study of hyperbolic systems. By using a Markov partition, the system can be made to resemble a discrete-time Markov process, with the long-term dynamical characteristics of the system represented as a Markov shift. The appellation 'Markov' is appropriate because the resulting dynamics of the system obeys the Markov property. The Markov partition thus allows standard techniques from symbolic dynamics to be applied, including the computation of expectation values, correlations, topological entropy, topological zeta functions, Fredholm determinants and the like. - In mathematics, the natural numbers are those used for counting (as in "there are "six" coins on the table") and ordering (as in "this is the "third" largest city in the country"). In common language, words used for counting are "cardinal numbers" and words used for ordering are "ordinal numbers". - In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members (also called "elements", or "terms"). The number of elements (possibly infinite) is called the "length" of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences) or the set of the first "n" natural numbers (for a sequence of finite length "n"). The position of an element in a sequence is its "rank" or "index"; it is the integer from which the element is the image. It depends on the context or of a specific convention, if the first element has index 0 or 1. When a symbol has been chosen for denoting a sequence, the "n"th element of the sequence is denoted by this symbol with "n" as subscript; for example, the "n"th element of the Fibonacci sequence is generally denoted "F". - Statistics is the study of the collection, analysis, interpretation, presentation, and organization of data. In applying statistics to, e.g., a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model process to be studied. Populations can be diverse topics such as "all people living in a country" or "every atom composing a crystal". Statistics deals with all aspects of data including the planning of data collection in terms of the design of surveys and experiments. - </math>| In probability theory and statistics, the Bernoulli distribution, named after Swiss scientist Jacob Bernoulli, is the probability distribution of a random variable which takes the value 1 with success probability of formula_7 and the value 0 with failure probability of formula_8. It can be used to represent a coin toss where 1 and 0 would represent "head" and "tail" (or vice versa), respectively. In particular, unfair coins would have formula_9. - Probability theory is the branch of mathematics concerned with probability, the analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single occurrences or evolve over time in an apparently random fashion. - An "a priori" probability is a probability that is derived purely by deductive reasoning. One way of deriving "a priori" probabilities is the principle of indifference, which has the character of saying that, if there are "N" mutually exclusive and exhaustive events and if they are equally likely, then the probability of a given event occurring is 1/"N". Similarly the probability of one of a given collection of "K" events is "K"/"N". - In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883. Given the information above, choose from the list below the object entity that exhibits the relation 'named after' with the subject 'bernoulli process'. Choices: - 1 - 2 - 5 - africa - air - aristotle - august - central - city - concept - jacob bernoulli - k - mathematics - point - stephen - technique - time - torus - translation - water
Answer:
jacob bernoulli