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Topic: Bernoulli process

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Bernoulli distribution

Bernoulli numbers

Bernoulli trial

Stochastic process

Probability

Likelihood principle

Halton sequences

Data clustering

Homoscedasticity

Bayesian probability

Monty Hall problem

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	Bernoulli process - Wikipedia, the free encyclopedia
		In probability and statistics, a Bernoulli process is a discrete-time stochastic process consisting of a sequence of independent random variables taking values over two letters.
		A Bernoulli process is a discrete-time stochastic process consisting of a finite or infinite sequence of independent random variables X
		The transfer operator, or Frobenius-Perron operator, of the Bernoulli map is solvable; the eigenvalues are multiples of 1/2, and the eigenfunctions are the Bernoulli polynomials.
	en.wikipedia.org /wiki/Bernoulli_process (579 words)

	Markov
		A discrete Weiner random process is a random walk process with p=q=0.5, and is known as binary white noise.
		A Brownian motion process, aka a continuous Weiner process or a Weiner-Levy process, is a random walk where the interval between consecutive values of the random sequence approaches zero.
		The process is a Markov process since (a) the current value of the process depends on the previous value, and (b) the magnitude of the change in the process is Gaussian with the change being +ve or -ve with equal probability.
	www.engr.udayton.edu /faculty/mdaniels/htm315/Markov.htm (1816 words)

	Bernoulli
		But for other applications it is easier to obtain a pressure reading at a given point rather than pulling out a ruler and measuring the area at a given point.
		Bernoulli's equation is only valid if one assumes the following: incompressible fluid (fluid velocity less than one third the speed of sound) and inviscid flow (this just means that the point in question along the flow is going to be away from where the flow and the object come into contact).
		Many believe that this explanation about the correlation of Bernoulli's principle and lift is incorrect because flat wings (such as seen on balsa wood airplanes, paper planes and others) also have managed to create lift.
	www.allstar.fiu.edu /aerojava/bernoulli.htm (1116 words)

	Sampling Distributions
		By population we mean a process in nature that has been reduced to numbers by measurement operations and then modeled as a probability distribution.
		Recall that in a Bernoulli process one outcome is called a success and the other a failure.
		This process of moving sideways across ideas and models from nature to measurement to probability distribution is called abduction.
	www.utah.edu /stat/materials/malloy/s-sampling-dis/index.html (6306 words)

	CACC
		The departure process of a queue in a queueing network is of special interest because it is the arrival process to other queues in the network.
		In this paper, the departure process of a discrete-time finite capacity queue, which is a special case, is also investigated.
		The generation function of the interdeparture process was fitted to a two-state MMBP by using a four moments matching technique.
	www.ece.ncsu.edu /cacc/show_techreport.php?id=174 (223 words)

	Introduction
		Essentially, the process is the mathematical abstraction of coin tossing, but because of its wide applicability, it is usually stated in terms of a sequence of generic trials that satisfy the following assumptions:
		As we noted earlier, the most obvious example of Bernoulli trials is coin tossing, where success means heads and failure means tails.
		Bernoulli trials are also formed when we sample from a dichotomous population.
	www.ds.unifi.it /VL/VL_EN/bernoulli/bernoulli1.html (997 words)

	The Poisson Process
		The Poisson process is one of the most important random processes in probability theory.
		It is widely used to model random "points" in time and space, such as the times of radioactive emissions, the arrival times of customers at a service center, and the positions of flaws in a piece of material.
		The process has a beautiful mathematical structure, and is used as a foundation for building a number of other, more complicated random processes.
	www.math.uah.edu /stat/poisson (94 words)

	Bernoulli scheme - Wikipedia, the free encyclopedia
		In mathematics, the Bernoulli scheme is a generalization of the Bernoulli process to more than two possible outcomes.
		The Bernoulli scheme is a stationary stochastic process, and, conversely, every stationary stochastic process is a Bernoulli scheme.
		Ornstein in 1968, states that two Bernoulli schemes with the same entropy are isomorphic.
	en.wikipedia.org /wiki/Bernoulli_scheme (384 words)

	Bernoulli trial
		In the theory of probability and statistics, a Bernoulli trial is an experiment whose outcome is random and can be either of two possible outcomes, called "success" and "failure." These last two words should not always be construed literally.
		Rolling a die, where for example we designate a six as "success" and everything else as a "failure".
		A Bernoulli process consists of repeatedly performing independent but identical Bernoulli trials, for instance flipping a coin 10 times.
	www.ebroadcast.com.au /lookup/encyclopedia/be/Bernoulli_trial.html (168 words)

	Stochastic Processes: Appendix A
		By counting process point of view, N(0) = 0 (the number of jumps in the process N is zero at t = 0).
		Karlin and Taylor ("A Second Course in Stochastic Processes", Academic Press, 1981), p.432, states "The general Lévy process can be represented as a sum of a Brownian motion, a uniform translation, and a limit (actually, an integral) of a one-parameter family of compound Poisson processes, where all the contributing basic processes are mutually independent".
		Bernoulli distribution is the building block to construct many other distributions in probability, such as binomial, geometric, hypergeometric, negative binomial, and Poisson.
	www.puc-rio.br /marco.ind/stoch-a.html (4265 words)

	Bernoulli Processes & The Binomial Distribution (Site not responding. Last check: 2007-10-13)
		A Bernoulli process is made up of a series of independent trials where each trial can result in either a "success" or a "failure." The probability of success is p on a given trial and p remains unchanged from trial to trial.
		If the assumptions of the Bernoulli process hold, we can find the probability that when we inspect 10 cars we will find the second and fifth car to have defects.
		When we sample from a Bernoulli process where p is the probability of a success, the probability of observing exactly r successes in n independent trials is called a binomial distribution.
	www.econ.utah.edu /~fowles/more_here.htm (225 words)

	EE126 Commentaries 11: Jean Walrand
		Roughly, a stochastic process is ergodic if statistics that do not depend on the initial phase of the process are constant.
		For instance, the Poisson process and the Brownian motion process are Markov.
		Note that this example shows that a function of a Markov process may not be a Markov process.
	robotics.eecs.berkeley.edu /~wlr/126/w11.htm (2705 words)

	Bernoulli Trials
		The Bernoulli trials process is one of the simplest, yet most important, of all random processes.
		The process consists of independent trials with two outcomes and with constant probabilities from trial to trial.
		The process leads to several important probability distributions: the binomial, geometric, and negative binomial.
	www.math.uah.edu /stat/bernoulli (69 words)

	Pólya's Urn and the Beta-Bernoulli Process (Site not responding. Last check: 2007-10-13)
		Pólya's urn scheme is a dichotomous sampling model that generalizes the hypergeometric model (sampling without replacement) and the Bernoulli model (sampling with replacement).
		Pólya's urn is one of the most famous examples of a random process in which the outcome variables are exchangeable, but dependent (in general).
		For a statistical application, suppose that we have a Bernoulli trials process (coin tosses for example) with an unknown probability of success.
	www.math.uah.edu.cob-web.org:8888 /stat/urn/Polya.xhtml (1355 words)

	Extra Problems for Chapter 4
		Thus the dependence between adjacent n-blocks of a stationary process does not grow linearly with n.
		be the entropy rate of the Y process (the sequence of coin tosses).
		Suppose we observe one of two stochastic processes but don't know which.
	www-isl.stanford.edu /~jat/eit2/webbook/exch4/exch4.html (225 words)

	Historical Analysis of MOT Closing Price Run Characteristics
		The length of a run would appear to indicate something about over-bought or oversold conditions, and so it might be of value as a sentiment indicator.
		That is, the next price in the series either will be up or down, so the problem of predicting the end of the run can be expressed in terms similar to those we might use to describe the probabilities of heads or tails when flipping a coin.
		Flipping a coin is a Bernoulli Process, because each flip is completely independent of what happened before.
	www.hybridtechnical.net /run-characterization/MOT.asp (704 words)

	Probability (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-10-13)
		Jakob Bernoullis Ars Conjectandi (posthumous, 1713) and Abraham de Moivres Doctrine of Chances (1718) treated the subject as a branch of mathematics.
		The theory of errors may be traced back to Roger Cotes's Opera Miscellanea (posthumous, 1722), but a memoir prepared by Thomas Simpson in 1755 (printed 1756) first applied the theory to the discussion of errors of observation.
		Daniel Bernoulli (1778) introduced the principle of the maximum product of the probabilities of a system of concurrent errors.
	probability.iqnaut.net.cob-web.org:8888 (2570 words)

	The multinomial distribution
		The use of the distribution is explained with an example.
		A multinomial trials process is a sequence of independent, identically distributed random variables
		Therefore this is a generalization of a Bernoulli trials process.
	www.cs.kuleuven.ac.be /~raf/homepage/publications/phd/node38.html (512 words)

	A Point Process Framework for Relating Neural Spiking Activity to Spiking History, Neural Ensemble, and Extrinsic ...
		In this model, the intensity (blue curve) was conditioned on the past spiking history, the spikes of 2 other neurons (neuron B, excitatory, red asterisks; neuron C, inhibitory, green asterisks), and on hand velocity.
		Past spiking history effect was modeled by a 120-order autoregressive process carrying a refractory period, recovery, and rebound excitation.
		Nonetheless, there remains some structure in the point process residual that is related to the hand velocity but was not captured by the velocity model.
	jn.physiology.org /cgi/content/full/93/2/1074 (7859 words)

	Agenda for December 7, 2005
		Bernoulli Processes = Independence of Successive Selections in a repetitive stochastic process (= sequences STDR REPS OK + probability)
		Bernoulli Processes and the BPDF and BCDF programs
		Note: Another technical phrase that has the same meaning as "sampling with replacement", i.e., "sampling with independent selections",etc., is the phrase Bernoulli Process.
	www.indiana.edu /~mathwhw/a118/a118_4058/agenda.12.07.html (442 words)

	Bernoulli Process (Site not responding. Last check: 2007-10-13)
		Also, you can right-click on the equation to bring up a control panel for changing the base font size for the equation.
		The probability of k successes in n trials of a Bernoulli process is given by the following formula:
		Here, the event E consists of outcomes where exactly k heads occur in n flips of a fair coin.
	web.mit.edu /webeq/currenthome/docs/tour/bernoulli.html (84 words)

	[No title]
		ª ¹óa¨Introduction (contd..)¨5State space of the process: Parameter index: ól¨Classification of processes* ¨qDiscrete vs. continuous state-space: Discrete vs. continuous parameter space: : Four types of processes: ón,¨+Discrete-state, discrete-parameter process ¨MExample: Number of cars in a service station, at the departure of each car.
		ób ¨-Discrete-state, continuous-parameter process ¨?Example: Number of cars in a service station at time t.
		¡L Móq.¨0Continuous-state, continuous-parameter process ¨KExample: Total service time of all the cars in the system, at time t.
	www.cse.uconn.edu /~ssg/cse221/lecture14.ppt (894 words)

	Probability - Wikipedia, the free encyclopedia (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-10-13)
		Christiaan Huygens (1657) gave the earliest known scientific treatment of the subject.
		See Ian Hacking's The Emergence of Probability for a history of the early development of the very concept of mathematical probability.
		The method of least squares is due to Adrien-Marie Legendre (1805), who introduced it in his Nouvelles méthodes pour la détermination des orbites des comètes (New Methods for Determining the Orbits of Comets).
	en.wikipedia.org.cob-web.org:8888 /wiki/Probability (2820 words)

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