Stochastic process conditional probability distribution

Conditional probability distributions were also computed for both sets of data. In principle, the dimensionality of an attractor can be estimated by noting the number of conditions required to yield a narrow (noise-limited) probability distribution.We observe a considerable narrowing of the probability distribution between the case of no conditions and that of one condition, but the number of data points is not sufficient to carry the process further. This narrowing is further evidence that the observed fluctuations are not purely stochastic in nature. 

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. 

For the Wiener process, we know that W(f) — W(5 ) is normally distributed with mean zero and variance f - this does not depend on whether there is additional information available regarding its value prior to min(f, s), that is to say, the Wiener process is a Markov process. More generally, when we say that a stochastic process is Markovian, we mean that the probability of a future event conditioned on the current state of the process and the past history of the process is the same as the probability conditioned on the current state of the process. Let A x), B(x) and Q(x) be three events dependent on state variable x, and let t+ > to > t- be three times, then, for a Markov process X(f) we have (using the notation for conditional probability) ... 

Ion chaimels are not always open. Single channel molecules switch suddenly (in less than one microsecond) between definite open and closed states. Single chaimel molecules open and close stochastically (in a process called spontaneous gating ) according to well-defined probability distributions [2] that are usually well described by Markov models (with a few rate constants that depend dramatically on conditions [6]). 

We first review fundamentals of the theory of stochastic processes. The system dynamics are specified by the set of its states, 5, and the transitions between them, S -> S, where S,S e 5. For example, the state S can denote the position of a Brownian particle, the numbers of molecules of different chemical species, or any other variable that characterizes the state of the system of interest. Here we restrict ourselves to processes for which the transition rates depend only on the system s instantaneous state, andnotontheentirety of its history. Such memoryless processes are known as Markovian and are applicable to a wide range of systems. We also assume that the transition rates do not explicitly depend on time, a condition known as stationarity. In this review we make the standard assumption that the transitions between the states are Poisson distributed random processes. In other words, the probability of transitioning from state S to state 5 in an infinitesimal interval, dt, is a S,S )dt, where a(S,S ) is the transition rate. 

A stochastic description of explosion phenomena is set up, both for isothermal and for exothermic reaction mechanisms. Numerical simulations and analytic study of the master equation show the appearence of long tail and multiple humps in the probability distribution, which subsist for a certain period of time. During this interval the system displays chaotic behavior, reflecting the random character of the ignition process. An estimate of the onset time of transient bimodality is carried out in terms of the size of the system, the intrinsic parameters, and the initial condition. The implications of the results in combustion are discussed. 

In Section 5.1 we introduce the stochastic processes. In Section 5.2 we will introduce Markov chains and define some terms associated with them. In Section 5.3 we find the n-step transition probability matrix in terms of one-step transition probability matrix for time invariant Markov chains with a finite state space. Then we investigate when a Markov ehain has a long-run distribution and discover the relationship between the long-run distribution of the Markov chain and the steady state equation. In Section 5.4 we classify the states of a Markov chain with a discrete state space, and find that all states in an irreducible Markov chain are of the same type. In Section 5.5 we investigate sampling from a Markov chain. In Section 5.6 we look at time-reversible Markov chains and discover the detailed balance conditions, which are needed to find a Markov chain with a given steady state distribution. In Section 5.7 we look at Markov chains with a continuous state space to determine the features analogous to those for discrete space Markov chains. 

The input F(x) in Equation 6.65 depends on the specifics of the ttansport process. For a given stochastic process of the general type given by Equation 6.50, or for a prescribed free energy landscape. Equation 6.65 is to be solved with appropriate boundary conditions. From such calculations, details about the probability distribution function and averages of the quantities associated with the translocation process can be obtained. We shall return to this calculational tool repeatedly for different experimental situations to be discussed in later chapters. 

Consider the stochastic translocation process given by Equation 10.39. Let mo be the number of monomers, which have been nucleated in the receiver compartment, to begin with. Starting from this initial condition, the probability distribution function of the first passage time r is given by Equation 6.86. Let us now consider the key results for the boundary conditions BCl and BC2 (Table 6.1). The results for the radiation boundary condition BCi can be obtained similarly by looking up the results in Section 6.7.4. 

For both independence and finite variance of the involved random variables, the central limit theorem holds a probability distribution gradually converges to the Gaussian shape. If the conditions of independence and finite variance of the random variables are not satisfied, other limit theorems must be considered. The study of limit theorems uses the concept of the basin of attraction of a probability distribution. All the probability density functions define a functional space. The Gaussian probability function is a fixed point attractor of stochastic processes in that functional space. The set of probability density functions that fulfill the requirements of the central limit theorem with independence and finite variance of random variables constitutes the basin of attraction of the Gaussian distribution. The Gaussian attractor is the most important attractor in the functional space, but other attractors also exist. 

There are various ways to classify mathematical models (5). First, according to the nature of the process, they can be classified as deterministic or stochastic. The former refers to a process in which each variable or parameter acquires a certain specific value or sets of values according to the operating conditions. In the latter, an element of uncertainty enters we cannot specify a certain value to a variable, but only a most probable one. Transport-based models are deterministic residence time distribution models in well-stirred tanks are stochastic. 

Relation (4.239) shows that k bubbles (bubbles having velocity v ) reach point x at time t + At because of the interaction with the other types of bubbles (the probability for this event is 1 — aAt) or because of the interaction with the composite liquid-solid medium (the probability for this event is aAt). At the same time, the bubbles that originate from the position x — v At without interaction with the nearly bubbles keep their velocity so the local distribution function of these individuals velocities is f (x, v, t). Due to the stochastic character of the described process, the transition probabilities from the state e to all k states verify the unification condition. Consequently, the probability p j will be written as... 

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