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Regenerative Process
In applied probability, a regenerative process is a class of stochastic process with the property that certain portions of the process can be treated as being statistically independent of each other. This property can be used in the derivation of theoretical properties of such processes. History Regenerative processes were first defined by Walter L. Smith in Proceedings of the Royal Society A in 1955. Definition A regenerative process is a stochastic process with time points at which, from a probabilistic point of view, the process restarts itself. These time point may themselves be determined by the evolution of the process. That is to say, the process is a regenerative process if there exist time points 0 ≤ ''T''0 < ''T''1 < ''T''2 < ... such that the post-''Tk'' process * has the same distribution as the post-''T''0 process * is independent of the pre-''Tk'' process for ''k ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Warehouse Md17
A warehouse is a building for storing goods. Warehouses are used by manufacturers, importers, exporters, wholesalers, transport businesses, customs, etc. They are usually large plain buildings in industrial parks on the outskirts of cities, towns, or villages. Warehouses usually have loading docks to load and unload goods from trucks. Sometimes warehouses are designed for the loading and unloading of goods directly from railways, airports, or seaports. They often have cranes and forklifts for moving goods, which are usually placed on ISO standard pallets and then loaded into pallet racks. Stored goods can include any raw materials, packing materials, spare parts, components, or finished goods associated with agriculture, manufacturing, and production. In India and Hong Kong, a warehouse may be referred to as a "godown". There are also godowns in the Shanghai Bund. History Prehistory and ancient history A warehouse can be defined functionally as a building in which to store bu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Applied Probability
Applied probability is the application of probability theory to statistical problems and other scientific and engineering domains. Scope Much research involving probability is done under the auspices of applied probability. However, while such research is motivated (to some degree) by applied problems, it is usually the mathematical aspects of the problems that are of most interest to researchers (as is typical of applied mathematics in general). Applied probabilists are particularly concerned with the application of stochastic processes, and probability more generally, to the natural, applied and social sciences, including biology, physics (including astronomy), chemistry, medicine, computer science and information technology, and economics. Another area of interest is in engineering: particularly in areas of uncertainty, risk management, probabilistic design, and Quality assurance. See also *Areas of application: **Ruin theory **Statistical physics **Stoichiometry and modelli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. Examples include the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. Stochastic processes have applications in many disciplines such as biology, chemistry, ecology, neuroscience, physics, image processing, signal processing, control theory, information theory, computer science, cryptography and telecommunications. Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance. Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include the Wiener process or Brownian motion process, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistically Independent
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two events are independent, statistically independent, or stochastically independent if, informally speaking, the occurrence of one does not affect the probability of occurrence of the other or, equivalently, does not affect the odds. Similarly, two random variables are independent if the realization of one does not affect the probability distribution of the other. When dealing with collections of more than two events, two notions of independence need to be distinguished. The events are called pairwise independent if any two events in the collection are independent of each other, while mutual independence (or collective independence) of events means, informally speaking, that each event is independent of any combination of other events in the collection. A similar notion exists for collections of random variables. Mutual independence implies pairwise independence, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wally Smith (mathematician)
Walter Laws Smith (November 12, 1926 – March 6, 2023) was a British-born American mathematician, known for his contributions to applied probability theory. Biography Smith was born in London on November 12, 1926. Smith received a B.A. in mathematics (1947) from Cambridge University, having gained First Class in the Mathematical Tripos Part 1 and Part 2. He then received an M.A. (1951) and Ph.D (1953) from Cambridge. His dissertation was entitled ''Stochastic Sequences of Events'' advised by Henry Daniels and David Cox (statistician), D. R. Cox, with whom he published the book ''Queues'' (1961) and also published with in his early years. He became a professor of statistics at The University of North Carolina Chapel Hill (1954–56 and 1958–), and he was an emeritus professor in the Department of Statistics and Operations Research. Smith was a fellow of the Institute of Mathematical Statistics, a fellow of the American Statistical Association (1966), a winner of the Adams Pri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of The Royal Society A
''Proceedings of the Royal Society'' is the main research journal of the Royal Society. The journal began in 1831 and was split into two series in 1905: * Series A: for papers in physical sciences and mathematics. * Series B: for papers in life sciences. Many landmark scientific discoveries are published in the Proceedings, making it one of the most historically significant science journals. The journal contains several articles written by the most celebrated names in science, such as Paul Dirac, Werner Heisenberg, Ernest Rutherford, Erwin Schrödinger, William Lawrence Bragg, Lord Kelvin, J.J. Thomson, James Clerk Maxwell, Dorothy Hodgkin and Stephen Hawking. In 2004, the Royal Society began ''The Journal of the Royal Society Interface'' for papers at the interface of physical sciences and life sciences. History The journal began in 1831 as a compilation of abstracts of papers in the ''Philosophical Transactions of the Royal Society'', the older Royal Society publication, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Independent And Identically Distributed
In 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. Introduction In statistics, we commonly deal with random samples. A random sample can be thought of as a set of objects that are chosen randomly. Or, more formally, it’s “a sequence of independent, identically distributed (IID) random variables”. In other words, the terms ''random sample'' and ''IID'' are basically one and the same. In statistics, we usually say “random sample,” but in probability it’s more common to say “IID.” * Identically Distributed means that there are no overall trends–the distribution doesn’t fluctuate and all items in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Renewal Process
Renewal theory is the branch of probability theory that generalizes the Poisson process for arbitrary holding times. Instead of exponentially distributed holding times, a renewal process may have any independent and identically distributed (IID) holding times that have finite mean. A renewal-reward process additionally has a random sequence of rewards incurred at each holding time, which are IID but need not be independent of the holding times. A renewal process has asymptotic properties analogous to the strong law of large numbers and central limit theorem. The renewal function m(t) (expected number of arrivals) and reward function g(t) (expected reward value) are of key importance in renewal theory. The renewal function satisfies a recursive integral equation, the renewal equation. The key renewal equation gives the limiting value of the convolution of m'(t) with a suitable non-negative function. The superposition of renewal processes can be studied as a special case of Markov ren ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Renewal Process
Renewal theory is the branch of probability theory that generalizes the Poisson process for arbitrary holding times. Instead of exponentially distributed holding times, a renewal process may have any independent and identically distributed (IID) holding times that have finite mean. A renewal-reward process additionally has a random sequence of rewards incurred at each holding time, which are IID but need not be independent of the holding times. A renewal process has asymptotic properties analogous to the strong law of large numbers and central limit theorem. The renewal function m(t) (expected number of arrivals) and reward function g(t) (expected reward value) are of key importance in renewal theory. The renewal function satisfies a recursive integral equation, the renewal equation. The key renewal equation gives the limiting value of the convolution of m'(t) with a suitable non-negative function. The superposition of renewal processes can be studied as a special case of Markov ren ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov Chain
A 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). It is named after the Russian mathematician Andrey Markov. Markov chains have many applications as statistical models of real-world processes, such as studying cruise control systems in motor vehicles, queues or lines of customers arriving at an airport, currency exchange rates and animal population dynamics. Markov processes are the basis for general stochastic simulation methods known as Markov chain Monte Carlo, which are used for simulating sampling from complex probability dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harris Chain
In the mathematical study of stochastic processes, a Harris chain is a Markov chain where the chain returns to a particular part of the state space an unbounded number of times. Harris chains are regenerative processes and are named after Theodore Harris. The theory of Harris chains and Harris recurrence is useful for treating Markov chains on general (possibly uncountably infinite) state spaces. Definition Let \ be a Markov chain on a general state space \Omega with stochastic kernel K. The kernel represents a generalized one-step transition probability law, so that P(X_\in C\mid X_n=x)=K(x,C) for all states x in \Omega and all measurable sets C\subseteq \Omega. The chain \ is a ''Harris chain''R. Durrett. ''Probability: Theory and Examples''. Thomson, 2005. . if there exists A\subseteq\Omega,\varepsilon>0, and probability measure \rho with \rho(\Omega)=1 such that # If \tau_A:=\inf \, then P(\tau_A 0, and let ''A'' and Ω be open sets containing ''x''0 and ''y''0 respec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reflected Brownian Motion
In probability theory, reflected Brownian motion (or regulated Brownian motion, both with the acronym RBM) is a Wiener process in a space with reflecting boundaries. In the physical literature, this process describes diffusion in a confined space and it is often called confined Brownian motion. For example it can describe the motion of hard spheres in water confined between two walls. RBMs have been shown to describe queueing models experiencing heavy traffic as first proposed by Kingman and proven by Iglehart and Whitt. Definition A ''d''–dimensional reflected Brownian motion ''Z'' is a stochastic process on \mathbb R^d_+ uniquely defined by * a ''d''–dimensional drift vector ''μ'' * a ''d''×''d'' non-singular covariance matrix ''Σ'' and * a ''d''×''d'' reflection matrix ''R''. where ''X''(''t'') is an unconstrained Brownian motion and ::Z(t) = X(t) + R Y(t) with ''Y''(''t'') a ''d''–dimensional vector where * ''Y'' is continuous and non–decreasing with ''Y''(0 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
