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02 Dec 14
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In probability theory, a stochastic (/stoʊˈkæstɪk/) process, or sometimes random process (widely used) is a collection of random variables, representing the evolution of some system of random values over time. This is the probabilistic counterpart to a deterministic process (or deterministic system). Instead of describing a process which can only evolve in one way (as in the case, for example, of solutions of an ordinary differential equation), in a stochastic or random process there is some indeterminacy: even if the initial condition (or starting point) is known, there are several (often infinitely many) directions in which the process may evolve.
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In probability theory, a stochastic (/stoʊˈkæstɪk/) process, or sometimes random process (widely used) is a collection of random variables, representing the evolution of some system of random values over time. This is the probabilistic counterpart to a deterministic process (or deterministic system). Instead of describing a process which can only evolve in one way (as in the case, for example, of solutions of an ordinary differential equation), in a stochastic or random process there is some indeterminacy: even if the initial condition (or starting point) is known, there are several (often infinitely many) directions in which the process may evolve.
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10 Nov 14
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One approach to stochastic processes treats them as functions of one or several deterministic arguments (inputs, in most cases regarded as time) whose values (outputs) are random variables: non-deterministic (single) quantities which have certain probability distributions
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Familiar examples of processes modeled as stochastic time series include stock market and exchange rate fluctuations, signals such as speech, audio and video, medical data such as a patient's EKG, EEG, blood pressure or temperature, and random movement such as Brownian motion or random walks. Examples of random fields include static images, random terrain (landscapes), wind waves or composition variations of a heterogeneous material
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14 Oct 14
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22 May 14
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there is some indeterminacy
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sequence of random variables and the time series associated with these random variables
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functions of one or several deterministic arguments (inputs, in most cases regarded as time) whose values (outputs) are random variables
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11 May 14
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stochastic processes treats them as functions of one or several deterministic arguments
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are random variables
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whose values
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13 Apr 14
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Instead of describing a process which can only evolve in one way (as in the case, for example, of solutions of an ordinary differential equation), in a stochastic or random process there is some indeterminacy: even if the initial condition (or starting point) is known, there are several (often infinitely many) directions in which the process may evolve.
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28 Dec 13
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25 Nov 13
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23 Nov 13
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in a stochastic or random process there is some indeterminacy: even if the initial condition (or starting point) is known, there are several (often infinitely many) directions in which the process may evolve.
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31 Aug 13
Dante-Gabryell Monson"In the simple case of discrete time, a stochastic process amounts to a sequence of random variables known as a time series (for example, see Markov chain). Another basic type of a stochastic process is a random field, whose domain is a region of space, in other words, a random function whose arguments are drawn from a range of continuously changing values. "
systems automenta intentional emergence synthesis netention stochastic
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16 Aug 13
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08 Jul 13
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This is the probabilistic counterpart to a deterministic process (or deterministic system). Instead of describing a process which can only evolve in one way (as in the case, for example, of solutions of an ordinary differential equation), in a stochastic or random process there is some indeterminacy: even if the initial condition (or starting point) is known, there are several (often infinitely many) directions in which the process may evolve.
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08 May 13
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collection of random variables; this is often used to represent the evolution of some random value, or system, over time
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discrete time, a stochastic process amounts to a sequence of random variables known as a time series
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14 Mar 13
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01 Feb 13
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06 Jan 13
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23 Oct 12
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15 May 12
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is a collection of random variables
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probabilistic counterpart to a deterministic process
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represent the evolution of some random value, or system, over time
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discrete time, a stochastic process amounts to a sequence of random variables known as a time series
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indexed by a totally ordered set T ("time")
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each
is an S-valued random variable -
S is then called the state space
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16 Apr 11
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). Instead of dealing with only one possible reality of how the process might evolve under time
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the counterpart to a deterministic process
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some indeterminacy in its future evolution described by probability distributions
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there are many possibilities the process might go to
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a sequence
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a region of space
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functions of one or several deterministic arguments
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values (outputs) are random variables
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13 Dec 10
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28 Apr 10
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In probability theory, a stochastic process, or sometimes random process, is the counterpart to a deterministic process (or deterministic system). Instead of dealing with only one possible reality of how the process might evolve under time (as is the case, for example, for solutions of an ordinary differential equation), in a stochastic or random process there is some indeterminacy in its future evolution described by probability distributions. This means that even if the initial condition (or starting point) is known, there are many possibilities the process might go to, but some paths may be more probable and others less.
In the simplest possible case (discrete time), a stochastic process amounts to a sequence of random variables known as a time series (for example, see Markov chain). Another basic type of a stochastic process is a random field, whose domain is a region of space, in other words, a random function whose arguments are drawn from a range of continuously changing values. One approach to stochastic processes treats them as functions of one or several deterministic arguments (inputs, in most cases regarded as time) whose values (outputs) are random variables: non-deterministic (single) quantities which have certain probability distributions. Random variables corresponding to various times (or points, in the case of random fields) may be completely different. The main requirement is that these different random quantities all have the same type.[1] Although the random values of a stochastic process at different times may be independent random variables, in most commonly considered situations they exhibit complicated statistical correlations.
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- What is the probability that each sample sequence is bounded?
- What is the probability that each sample sequence is monotonic?
- What is the probability that each sample sequence has a limit as the index approaches ∞?
- What is the probability that the series obtained from a sample sequence from f(i) converges?
- What is the probability distribution of the sum?
- What is the probability that it is bounded/integrable/continuous/differentiable...?
- What is the probability that it has a limit at ∞
- What is the probability distribution of the integral?
Examples
The paradigm of continuous stochastic process is that of the Wiener process. In its original form the problem was concerned with a particle floating on a liquid surface, receiving "kicks" from the molecules of the liquid. The particle is then viewed as being subject to a random force which, since the molecules are very small and very close together, is treated as being continuous and, since the particle is constrained to the surface of the liquid by surface tension, is at each point in time a vector parallel to the surface. Thus the random force is described by a two component stochastic process; two real-valued random variables are associated to each point in the index set, time, (note that since the liquid is viewed as being homogeneous the force is independent of the spatial coordinates) with the domain of the two random variables being R, giving the x and y components of the force. A treatment of Brownian motion generally also includes the effect of viscosity, resulting in an equation of motion known as the Langevin equation.
If the index set of the process is N (the natural numbers), and the range is R (the real numbers), there are some natural questions to ask about the sample sequences of a process {Xi}i ∈ N, where a sample sequence is {X(ω)i}i ∈ N.
Similarly, if the index space I is a finite or infinite interval, we can ask about the sample paths {X(ω)t}t ∈ I
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18 Oct 09
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in a stochastic or random process there is some indeterminacy in its future evolution described by probability distributions. This means that even if the initial condition (or starting point) is known, there are many possibilities the process might go to, but some paths are more probable and others less.
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26 Jun 09
Fernando Sánchez ZamoraA stochastic process, or sometimes random process, is the counterpart to a deterministic process (or deterministic system) in probability theory. Instead of dealing with only one possible "reality" of how the process might evolve under time (as is the cas
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06 Aug 08
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A stochastic process, or sometimes random process, is the counterpart to a deterministic process
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21 Mar 08
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26 May 07
is an S-valued random variable
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