Probability function joint

Regarding Eq. (6.3), note that we did not say anything about the joint probability function. While it seems intuitively clear that such function exists, its evaluation involves analysis of the time evolution of the system. To see this more clearly let us focus on classical mechanics, and recall that the observables A and B correspond to dynamical variables A and B that are function of positions and momenta of all particles in the system... 

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... 

The probabilities defined by Eqs. (33.26) and (33.27), can be calculated by integrating the joint probability function of the significant random variables using Monte Carlo or FORM numeric methods (first order reliability methods). A description of these methods can be found in Refs. 11 and 12. 

The purpose of this paper is to present a synthetic picture of the work done up to now concerning this "modelling". We will first explain the basic principles, and we will show that, for a multireactive mixture, the joint probability function of the species concentrations is a key quantity. Secondly, we intend to briefly review the three groups of models that are presently under investigation. Finally, we will emphasize the weakest point of all these models, the one which has to attract all our efforts in the future. 

The cluster averages appearing in (38) or (47) are performed with the aid of a joint probablity function... 

The joint probability function for two-, three-,etc., site clusters needed in the T- matrix averaging procedure can easily be computed. The matrix has less symmetry than in the case of uncorrelated sites, since the clusters averaged over do not occur in symmetric pairs, as in the case when the joint probability P is given by a produce of independent random variables. Specifically, does not have to equal... 

If this criterion is based on the maximum-likelihood principle, it leads to those parameter values that make the experimental observations appear most likely when taken as a whole. The likelihood function is defined as the joint probability of the observed values of the variables for any set of true values of the variables, model parameters, and error variances. The best estimates of the model parameters and of the true values of the measured variables are those which maximize this likelihood function with a normal distribution assumed for the experimental errors. 

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... 

Equation (3-104) (sometimes called the stationarity property of a probability density function) follows from the definition of the joint distribution function upon making the change of variable t = t + r... 

The physical interpretation of these joint moments is similar in every respect to the interpretation already given for moments of the form ak = E[k]. Thus, a . .. provides a measure of the center of mass of the joint probability density function p 1,...,second order central moments provide a measure of the spread of this density function about its center of mass.30... 

The joint characteristic function is thus seen to be the -dimensional Fourier transform of the joint probability density function The -dimensional Fourier transform, like its one-dimensional counterpart, can be inverted by means of the formula... 

In other words, knowledge of the joint characteristic function of a family of random variables is tantamount to knowledge of their joint probability density function and vice versa. 

This conditional probability can also be written in terms of the joint probability density function for lt , n+m as follows ... 

The conditional probability density functions defined by Eq. (3-170) are joint probability density functions for fixed values of xn... 

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... 

In terms of these functions, we define all possible joint probability density functions for a time function X(t) by writing... 

Joint distribution functions, in terms of associated probability density functions, 133 notation, 143... 

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,... 

If we now further assume that measurements from different experiments are independent, the joint probability density function for the all the measured responses is simply the product,... 

The Loglikelihood function is the log of the joint probability density function and is regarded as a function of the parameters conditional on the observed responses. Hence, we have... 

One more quantitative way to characterize the chemical structure of copolymers is based on the consideration of chemical correlation functions (correlators) [2]. The simplest of these, Ya k), describes the joint probability of finding two randomly chosen monomeric units divided along the macromolecule by an arbitrary sequence Uk ... 

The correlator (6) is of the utmost importance because its generating function enters into an expression which describes the angular dependence of intensity of scattering of light or neutrons [3]. It is natural to extend expression (6) for the two-point chemical correlation function by introducing the w-point correlator ya1... (kl...,kn l) which equals the joint probability of finding in a macromolecule n monomeric units Maj.Ma> divided by (n-1) arbitrary sequences... 

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-... 

Solutions to the Schrodinger equation Hcj) = E(f> are the molecular wave functions 0, that describe the entangled motion of the three particles such that (j) 4> represents the density of protons and electron as a joint probability without any suggestion of structure. Any other molecular problem, irrespective of complexity can also be developed to this point. No further progress is possible unless electronic and nuclear variables are separated via the adiabatic simplification. In the case of Hj that means clamping the nuclei at a distance R apart to generate a Schrodinger equation for electronic motion only, in atomic units,... 

Even for the resonant transmission through the Sinai billiard, computations show that many eigenfunctions contribute to the scattering wave function as shown in fig. 1. An assumption of a complex RGF for the scattering function (9) means that the joint probability density has the form... 

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