Joint probability distribution function

Giacovazzo, C. and SUiqi, D. (2001). The method of joint probability distribution functions apphed to MAD techniques. The two-wavelength case for acentric crystals. Acta Crystallogr. A 57, 700-707. 

A physical system S that evolves probabilistically in time can be mathematically described by a time-dependent random variable X(t). It is assumed that (1) one can measure values xlt x2, x3,.. . , xn of X(t) at instants ti,t2,ti,...,tn representing the possible states of the system S and (2) one can define a set of joint probability distribution functions... 

Joint probability distribution function of the local velocity and acceleration, Eq. (44)... 

Although the normal mode representation is very useful, the physically observable coordinate is usually the system coordinate q. In the Langevin representation of the dynamics, the trajectory q(t) is characterized by a random force which affects its motion. In the equivalent Fokker-Planck representation of Langevin dynamics, all dynamical information lies in the joint probability distribution function that the particle at time t has position and velocity q, v, given that at time r = 0 its position and velocity were q, v. ... 

A final important point is that all of the above has only been for isothermal conditions in single-phase systems. Extension to other cases requires the introduction of interactions with the second phase and/or heat exchange walls, and so on, and the age-distribution and micromixing functions depend nnuch more on the details of the system. A formal treatment would use joint probability distribution functions, but this rapidly gets extremely complex. The population balance models can give some insight into the two-phase situation, and will be discussed below. 

Ferreira and Soares describe the joint probability distribution function of longterm hydraulic conditions. Especially when the main interest is in the design of flood defence structures, the extreme conditions are important, which implies that the dependence between hydraulic conditions needs to be accounted for. The joint probability analysis of extreme waves and water levels thus is significant in order to estimate more accurately the extreme environmental loading on a coastal structure. Because wind setup (storm surge) and wave conditions depend on the same driving force, a strong dependence between them is observed under extreme conditions. 

The properties of joint distribution functions can be stated most easily in terms of their associated probability density functions. The n + mth order joint probability density function px. . , ( > ) is defined by the equation... 

It can be shown that the right-hand side of Eq. (3-208) is the -dimensional characteristic function of a -dimensional distribution function, and that the -dimensional distribution function of afn, , s n approaches this distribution function. Under suitable additional hypothesis, it can also be shown that the joint probability density function of s , , sjn approaches the joint probability density function whose characteristic function is given by the right-hand side of Eq. (3-208). To preserve the analogy with the one-dimensional case, this distribution (density) function is called the -dimensional, zero mean gaussian distribution (density) function. The explicit form of this density function can be obtained by taking the i-dimensional Fourier transform of e HsA, with the result.45... 

If the mathematical model of the process under consideration is adequate, it is very reasonable to assume that the measured responses from the i,h experiment are normally distributed. In particular the joint probability density function conditional on the value of the parameters (k and ,) is of the form,... 

The knowledge required to implement Bayes formula is daunting in that a priori as well as class conditional probabilities must be known. Some reduction in requirements can be accomplished by using joint probability distributions in place of the a priori and class conditional probabilities. Even with this simplification, few interpretation problems are so well posed that the information needed is available. It is possible to employ the Bayesian approach by estimating the unknown probabilities and probability density functions from exemplar patterns that are believed to be representative of the problem under investigation. This approach, however, implies supervised learning where the correct class label for each exemplar is known. The ability to perform data interpretation is determined by the quality of the estimates of the underlying probability distributions. 

We can, therefore, let /cx be the subject of our calculations (which we approximate via an array in the computer). Post-simulation, we desire to examine the joint probability distribution p(N, U) at normal thermodynamic conditions. The reweighting ensemble which is appropriate to fluctuations in N and U is the grand-canonical ensemble consequently, we must specify a chemical potential and temperature to determine p. Assuming -7CX has converged upon the true function In f2ex, the state probabilities are given by... 

The importance of chemical-reaction kinetics and the interaction of the latter with transport phenomena is the central theme of the contribution of Fox from Iowa State University. The chapter combines the clarity of a tutorial with the presentation of very recent results. Starting from simple chemistry and singlephase flow the reader is lead towards complex chemistry and two-phase flow. The issue of SGS modeling discussed already in Chapter 2 is now discussed with respect to the concentration fields. A detailed presentation of the joint Probability Density Function (PDF) method is given. The latter allows to account for the interaction between chemistry and physics. Results on impinging jet reactors are shown. When dealing with particulate systems a particle size distribution (PSD) and corresponding population balance equations are intro-... 

The wave function P contains all information of the joint probability distribution of the electrons. For example, the two-electron density is obtained from the wave function by integration over the spin and space coordinates of all but two electrons. It describes the joint probability of finding electron 1 at r, and electron 2 at r2. The two-electron density cannot be obtained from elastic Bragg scattering. 

The expression (2.1) is not the precise number of reactive collisions, but the average. The actual number fluctuates around it and we want to find the resulting fluctuations in the rij around the macroscopic values determined by (2.2). In order to describe them one needs the joint probability distribution P(n, t). Although it is written as a function of all rij, it is defined on the accessible sublattice alone. Alternatively one may regard it as a distribution over the whole physical octant, which is zero on all points that are not accessible. 

Prompted by these considerations, Gillespie [388] introduced the reaction probability density function p (x, l), which is a joint probability distribution on the space of the continuous variable x (0 < x < oc) and the discrete variable l (1 = 1,..., to0). This function is used as p (x, l) Ax to define the probability that given the state n(t) at time t, the next event will occur in the infinitesimal time interval (t + x,t + x + Ax), AND will be an Ri event. Our first step toward finding a legitimate method for assigning numerical values to x and l is to derive, from the elementary conditional probability hi At, an analytical expression for p (x, l). To this end, we now calculate the probability p (x, l) Ax as the product po (x), the probability at time t that no event will occur in the time interval (t, t + x) TIMES a/ Ax, the subsequent probability that an R.i... 

Collins, D. M. Extrapolative filtering. I. Maximization of resolution for onedimensional positive density functions. Acta Cryst. A34, 533-541 (1978). Bricogne, G. A Bayesian statistical theory of the phase problem. I. A multichannel maximum-entropy formalism for constructing generalized joint probability distributions of structure factors. Acta Cryst. A44, 517-545 (1988). 

Demonstrate that the probability distribution function of the end-to-end distance i of a freely jointed chain can be expressed in terms of the inverse Langevin function C (x ) of the ratio x = RjR y of the end-to-end distance R to its maximum value i max =... 

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. 

As we have already discussed in Section II, another characteristic of the quantum field is its phase distribution. The phase distribution of the quantum field can be calculated from the quasidistribution functions by integrating over the radial variable. In this way we get a kind of phase distribution that can be considered as an approximate description of the phase properties of the field. One can calculate the s-parametrized phase distributions, corresponding to the 5-parametrized quasidistributions, for particular quantum states of the field [16]. However, a better way to study quantum phase properties is to use the Hermitian phase formalism introduced by Pegg and Barnett [11-13]. We have already introduced this formalism in Section II. Now, we apply this formalism to study the evolution of the phase properties of the two modes in the SHG process. In this case we have a two-mode field which requires a modification of the formulas presented in Section II into a two-mode case. The modification is rather trivial, and for the joint probability distribution for the continuous phase variables 0 and 0/, describing phases of the two modes, we get the formula [53]... 

The joint probability distribution given by Eq. (143) can be evaluated numerically and an example of such distribution is shown in Fig. 13, where the function P(Qa, 0/>) is plotted for several values of the scaled time x and the initial mean number of photons of the fundamental mode Na = 10. Initially the... 

Notice that two assumptions have been made normality of the responses and constant variance. The result is that the conditional distribution itself is normally distributed with mean 0O + (fix and variance joint distribution function at any level of X can be sliced and still have a normal distribution. Also, any conditional probability distribution function of Y has the same standard deviation after scaling the resulting probability distribution function to have an area of 1. 

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