Independent and identically distributed random variablesIn probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usually abbreviated as i.i.d., iid, or IID. IID was first defined in statistics and finds application in different fields such as data mining and signal processing. Statistics commonly deals with random samples. A random sample can be thought of as a set of objects that are chosen randomly.
Coin flippingCoin flipping, coin tossing, or heads or tails is the practice of throwing a coin in the air and checking which side is showing when it lands, in order to choose between two alternatives, heads or tails, sometimes used to resolve a dispute between two parties. It is a form of sortition which inherently has two possible outcomes. The party who calls the side that is facing up when the coin lands wins. Coin flipping was known to the Romans as navia aut caput ("ship or head"), as some coins had a ship on one side and the head of the emperor on the other.
Ornstein isomorphism theoremIn mathematics, the Ornstein isomorphism theorem is a deep result in ergodic theory. It states that if two 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, including Markov chains and subshifts of finite type, Anosov flows and Sinai's billiards, ergodic automorphisms of the n-torus, and the continued fraction transform.
Exchangeable random variablesIn statistics, an exchangeable sequence of random variables (also sometimes interchangeable) is a sequence X1, X2, X3, ... (which may be finitely or infinitely long) whose joint probability distribution does not change when the positions in the sequence in which finitely many of them appear are altered. Thus, for example the sequences both have the same joint probability distribution. It is closely related to the use of independent and identically distributed random variables in statistical models.
Fair coinIn probability theory and statistics, a sequence of independent Bernoulli trials with probability 1/2 of success on each trial is metaphorically called a fair coin. One for which the probability is not 1/2 is called a biased or unfair coin. In theoretical studies, the assumption that a coin is fair is often made by referring to an ideal coin. John Edmund Kerrich performed experiments in coin flipping and found that a coin made from a wooden disk about the size of a crown and coated on one side with lead landed heads (wooden side up) 679 times out of 1000.
Law of large numbersIn probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value and tends to become closer to the expected value as more trials are performed. The LLN is important because it guarantees stable long-term results for the averages of some random events.
Transfer operatorIn mathematics, the transfer operator encodes information about an iterated map and is frequently used to study the behavior of dynamical systems, statistical mechanics, quantum chaos and fractals. In all usual cases, the largest eigenvalue is 1, and the corresponding eigenvector is the invariant measure of the system. The transfer operator is sometimes called the Ruelle operator, after David Ruelle, or the Perron–Frobenius operator or Ruelle–Perron–Frobenius operator, in reference to the applicability of the Perron–Frobenius theorem to the determination of the eigenvalues of the operator.
