Stochastic processes probability distribution

Especially in the process industries various stochastic methods can be applied to cope with random demand. In many cases, random demands can be described by probability distributions, the parameters of which may be estimated from history. This is not always possible, the car industry is an example. No two cars are exactly the same and after a few years there is always a new model which may change the demand pattern significantly. 

Detailed modeling study of practical sprays has a fairly short history due to the complexity of the physical processes involved. As reviewed by O Rourke and Amsden, 3l() two primary approaches have been developed and applied to modeling of physical phenomena in sprays (a) spray equation approach and (b) stochastic particle approach. The first step toward modeling sprays was taken when a statistical formulation was proposed for spray analysis. 541 Even with this simplification, however, the mathematical problem was formidable and could be analyzed only when very restrictive assumptions were made. This is because the statistical formulation required the solution of the spray equation determining the evolution of the probability distribution function of droplet locations, sizes, velocities, and temperatures. The spray equation resembles the Boltzmann equation of gas dynamics[542] but has more independent variables and more complex terms on its right-hand side representing the effects of nucleations, collisions, and breakups of droplets. 

In this section, we begin the description of Brownian motion in terms of stochastic process. Here, we establish the link between stochastic processes and diffusion equations by giving expressions for the drift velocity and diffusivity of a stochastic process whose probability distribution obeys a desired diffusion equation. The drift velocity vector and diffusivity tensor are defined here as statistical properties of a stochastic process, which are proportional to the first and second moments of random changes in coordinates over a short time period, respectively. In Section VILA, we describe Brownian motion as a random walk of the soft generalized coordinates, and in Section VII.B as a constrained random walk of the Cartesian bead positions. 

The classical, frequentist approach in statistics requires the concept of the sampling distribution of an estimator. In classical statistics, a data set is commonly treated as a random sample from a population. Of course, in some situations the data actually have been collected according to a probability-sampling scheme. Whether that is the case or not, processes generating the data will be snbject to stochastic-ity and variation, which is a sonrce of uncertainty in nse of the data. Therefore, sampling concepts may be invoked in order to provide a model that accounts for the random processes, and that will lead to confidence intervals or standard errors. The population may or may not be conceived as a finite set of individnals. In some situations, such as when forecasting a fnture value, a continuous probability distribution plays the role of the popnlation. 

For any r-component stochastic process one may ignore a number of components and the remaining s components again constitute a stochastic process. But, if the r-component process is Markovian, the process formed by the sfirst example above each velocity component is itself Markovian in chemical reactions, however, the future probability distribution of the amount of each chemical component is determined by the present amounts of all components. 

Next let us show how one can compute the proteasome output if the transport rates are given. In our model we assume that the proteasome has a single channel for the entry of the substrate with two cleavage centers present at the same distance from the ends, yielding in a symmetric structure as confirmed by experimental studies of its structure. In reality a proteasome has six cleavage sites spatially distributed around its central channel. However, due to the geometry of its locations, we believe that a translocated protein meets only two of them. Whether the strand is indeed transported or cleaved at a particular position is a stochastic process with certain probabilities (see Fig. 14.5). 

Here, X. is the stochastic state vector, B(r,X.j) is a vector describing the contribution of the diffusion to the stochastic process and W. is a vector with the same dimensions as X. and B(t,X.j). After Eqs. (4.94) and (4.95), the W,. vector is a Wiener process (we recall that this process is stochastic with a mean value equal to zero and a gaussian probability distribution) with the same dimensions as D(t,X,) ... 

Chvosta et alP consider another case where the work probability distribution function can be determined. They study a two energy-level system, modelled as a stochastic, Markovian process, where the transition rates and energies depend on time. Like the previous examples it provides an exact model that can be used to assist in identifying the accuracy of approximate, numerical studies. Ge and Qian extended the stochastic derivation for a Markovian chain to a inhomogeneous Markov chain. 

We can measure and discuss z(Z) directly, keeping in mind that we will obtain different realizations (stochastic trajectories) of this function from different experiments performed imder identical conditions. Alternatively, we can characterize the process using the probability distributions associated with it. P(z, Z)random variable z at time Z is in the interval between z and z +- dz. P2(z2t2 zi fi )dzidz2 is the probability that z will have a value between zi and zi + dz at Zi and between Z2 and Z2 -F t/z2 at t, etc. The time evolution of the process, if recorded in times Zo, Zi, Z2, - - , Zn is most generally represented by the joint probability distribution Piz t , , z iUp. Note that any such joint distribution function can be expressed as a reduced higher-order function, for example. 

If x(Z) is real then = x . Equation (7.69) resolves x(Z) into its spectral components, and associates with it a set of coefficients x such that x p is the strength or intensity of the spectral component of frequency However, since each realization of x(Z) in the interval 0,..., T yields a different set x , the variables x are themselves random, and characterized by some (joint) probability function P( x ). This distribution in turn is characterized by its moments, and these can be related to properties of the stochastic process x(Z). For example, the averages x satisfy... 

We have already noted the difference between the Langevin description of stochastic processes in terms of the stochastic variables, and the master or Fokker-Planck equations that focus on their probabilities. Still, these descriptions are equivalent to each other when applied to the same process and variables. It should be possible to extract information on the dynamics of stochastic variables from the time evolution of their probabihty distribution, for example, the Fokker-Planck equation. Here we show that this is indeed so by addressing the passage time distribution associated with a given stochastic process. In particular we will see (problem 14.3) that the first moment of this distribution, the mean first passage time, is very useful for calculating rates. 

A series of probable transitions between states can be described with the Markov chain. A Markovian stochastic process is memoryless, and this is illustrated subsequently. We generate a sequence of random variables, (yo, yi, yi, ), so that each time t > 0, the next state yt+i would be sampled from a distribution P(y,+ily,), which would depend only on the current state of the chain, y,. Thus, given y, the next state y,+i would not depend additionally on the history of the chain (yo, yi, yi,---, y i). The name Markov chain is used to describe this sequence, and the transition kernel of the chain is i (.l.) does not depend on t if we assume that the chain is time homogeneous. A detail description of the Markov model is provided in Chapter 26. 

For a coherent state oto), p = a0)(oto, and the quasiprobability distribution P a) = 8 (a — oto) giving ((a+)man) = (a )man). When P(a) is a well-behaved, positive definite function, it can be considered as a probability distribution function of a classical stochastic process, and the field with such a P function is said to have classical analog. However, the P function can be highly singular or can take negative values, in which case it does not satisfy requirements for the probability distribution, and the field states with such a P function are referred to as nonclassical states. 

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