Convergence of random variablesIn probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to statistics and stochastic processes. The same concepts are known in more general mathematics as stochastic convergence and they formalize the idea that a sequence of essentially random or unpredictable events can sometimes be expected to settle down into a behavior that is essentially unchanging when items far enough into the sequence are studied.
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.
Random variableA random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. The term 'random variable' can be misleading as it is not actually random nor a variable, but rather it is a function from possible outcomes (e.g., the possible upper sides of a flipped coin such as heads and tails ) in a sample space (e.g., the set ) to a measurable space (e.g., in which 1 corresponding to and −1 corresponding to ), often to the real numbers.
Complex random variableIn probability theory and statistics, complex random variables are a generalization of real-valued random variables to complex numbers, i.e. the possible values a complex random variable may take are complex numbers. Complex random variables can always be considered as pairs of real random variables: their real and imaginary parts. Therefore, the distribution of one complex random variable may be interpreted as the joint distribution of two real random variables.
Markov chainA Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happens next depends only on the state of affairs now." A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC). A continuous-time process is called a continuous-time Markov chain (CTMC).
Continuous uniform distributionIn probability theory and statistics, the continuous uniform distributions or rectangular distributions are a family of symmetric probability distributions. Such a distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds. The bounds are defined by the parameters, and which are the minimum and maximum values. The interval can either be closed (i.e. ) or open (i.e. ). Therefore, the distribution is often abbreviated where stands for uniform distribution.
Supply chainA supply chain, sometimes expressed as a "supply-chain", is a complex logistics system that consists of facilities that convert raw materials into finished products and distribute them to end consumers or end customers. Meanwhile, supply chain management deals with the flow of goods within the supply chain in the most efficient manner. In sophisticated supply chain systems, used products may re-enter the supply chain at any point where residual value is recyclable. Supply chains link value chains.
Discrete uniform distributionIn probability theory and statistics, the discrete uniform distribution is a symmetric probability distribution wherein a finite number of values are equally likely to be observed; every one of n values has equal probability 1/n. Another way of saying "discrete uniform distribution" would be "a known, finite number of outcomes equally likely to happen". A simple example of the discrete uniform distribution is throwing a fair die. The possible values are 1, 2, 3, 4, 5, 6, and each time the die is thrown the probability of a given score is 1/6.
Value chainA value chain is a progression of activities that a firm operating in a specific industry performs in order to deliver a valuable product (i.e., good and/or service) to the end customer. The concept comes through business management and was first described by Michael Porter in his 1985 best-seller, Competitive Advantage: Creating and Sustaining Superior Performance. The idea of the value chain is based on the process view of organizations, the idea of seeing a manufacturing (or service) organization as a system, made up of subsystems each with inputs, transformation processes and outputs.
Product (business)In marketing, a product is an object, or system, or service made available for consumer use as of the consumer demand; it is anything that can be offered to a market to satisfy the desire or need of a customer. In retailing, products are often referred to as merchandise, and in manufacturing, products are bought as raw materials and then sold as finished goods. A service is also regarded as a type of product. In project management, products are the formal definition of the project deliverables that make up or contribute to delivering the objectives of the project.
Monotone convergence theoremIn the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the convergence of monotonic sequences (sequences that are non-decreasing or non-increasing) that are also bounded. Informally, the theorems state that if a sequence is increasing and bounded above by a supremum, then the sequence will converge to the supremum; in the same way, if a sequence is decreasing and is bounded below by an infimum, it will converge to the infimum.
Markov modelIn probability theory, a Markov model is a stochastic model used to model pseudo-randomly changing systems. It is assumed that future states depend only on the current state, not on the events that occurred before it (that is, it assumes the Markov property). Generally, this assumption enables reasoning and computation with the model that would otherwise be intractable. For this reason, in the fields of predictive modelling and probabilistic forecasting, it is desirable for a given model to exhibit the Markov property.
Supply chain managementIn commerce, supply chain management (SCM) deals with a system of procurement (purchasing raw materials/components), operations management (ensuring the production of high-quality products at high speed with good flexibility and low production cost), logistics and marketing channels, so that the raw materials can be converted into a finished product and delivered to the end customer.
Markov chain Monte CarloIn statistics, Markov chain Monte Carlo (MCMC) methods comprise a class of algorithms for sampling from a probability distribution. By constructing a Markov chain that has the desired distribution as its equilibrium distribution, one can obtain a sample of the desired distribution by recording states from the chain. The more steps that are included, the more closely the distribution of the sample matches the actual desired distribution. Various algorithms exist for constructing chains, including the Metropolis–Hastings algorithm.
Markov propertyIn probability theory and statistics, the term Markov property refers to the memoryless property of a stochastic process, which means that its future evolution is independent of its history. It is named after the Russian mathematician Andrey Markov. The term strong Markov property is similar to the Markov property, except that the meaning of "present" is defined in terms of a random variable known as a stopping time. The term Markov assumption is used to describe a model where the Markov property is assumed to hold, such as a hidden Markov model.
Hidden Markov modelA hidden Markov model (HMM) is a statistical Markov model in which the system being modeled is assumed to be a Markov process — call it — with unobservable ("hidden") states. As part of the definition, HMM requires that there be an observable process whose outcomes are "influenced" by the outcomes of in a known way.
Uniform convergenceIn the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions converges uniformly to a limiting function on a set as the function domain if, given any arbitrarily small positive number , a number can be found such that each of the functions differs from by no more than at every point in .
Limit of a sequenceAs the positive integer becomes larger and larger, the value becomes arbitrarily close to . We say that "the limit of the sequence equals ." In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the symbol (e.g., ). If such a limit exists, the sequence is called convergent. A sequence that does not converge is said to be divergent. The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests.
Lévy's continuity theoremIn probability theory, Lévy’s continuity theorem, or Lévy's convergence theorem, named after the French mathematician Paul Lévy, connects convergence in distribution of the sequence of random variables with pointwise convergence of their characteristic functions. This theorem is the basis for one approach to prove the central limit theorem and is one of the major theorems concerning characteristic functions. Suppose we have If the sequence of characteristic functions converges pointwise to some function then the following statements become equivalent: Rigorous proofs of this theorem are available.
Sequence alignmentIn bioinformatics, a sequence alignment is a way of arranging the sequences of DNA, RNA, or protein to identify regions of similarity that may be a consequence of functional, structural, or evolutionary relationships between the sequences. Aligned sequences of nucleotide or amino acid residues are typically represented as rows within a matrix. Gaps are inserted between the residues so that identical or similar characters are aligned in successive columns.
