Joint-probability density

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... 

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... 

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... 

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-... 

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... 

A theoretical framework based on the one-point, one-time joint probability density function (PDF) is developed. It is shown that all commonly employed models for turbulent reacting flows can be formulated in terms of the joint PDF of the chemical species and enthalpy. Models based on direct closures for the chemical source term as well as transported PDF methods, are covered in detail. An introduction to the theory of turbulence and turbulent scalar transport is provided for completeness. 

In order to compare various reacting-flow models, it is necessary to present them all in the same conceptual framework. In this book, a statistical approach based on the one-point, one-time joint probability density function (PDF) has been chosen as the common theoretical framework. A similar approach can be taken to describe turbulent flows (Pope 2000). This choice was made due to the fact that nearly all CFD models currently in use for turbulent reacting flows can be expressed in terms of quantities derived from a joint PDF (e.g., low-order moments, conditional moments, conditional PDF, etc.). Ample introductory material on PDF methods is provided for readers unfamiliar with the subject area. Additional discussion on the application of PDF methods in turbulence can be found in Pope (2000). Some previous exposure to engineering statistics or elementary probability theory should suffice for understanding most of the material presented in this book. 

In this book, an alternative description based on the joint probability density function (PDF) of the species concentrations will be developed. (Exact definitions of the joint PDF and related quantities are given in Chapter 3.) The RTD function is in fact the PDF of the fluid-element ages as they leave the reactor. The relationship between the PDF description and the RTD function can be made transparent by defining a fictitious chemical species... 

Second Order Stationarity. With only a single realization of the random function it would be impossible to make any meaningful inferences about the random function if we did not make some assumptions about its stationarity. A random function is said to be strictly stationary if the joint probability density function for k arbitrary points is invariant under simultaneous translation of all... 

The conditional probability densities are related to the joint probability densities by the relation... 

A complete probabilistic description of the system is obtained either by specifying all of the joint probability densities, pr, r = 1, 2, 3,..., or by specifying the singlet probability density p2 and all of the conditional... 

Since the joint probability density of the complete set of positions and momenta is Gaussian, the distribution of any subset must also be Gaussian. But the characteristic function corresponding to a Gaussian density takes a simple form, and in the present case is... 

Exercise. What is the form of the joint probability density if all variables are mutually independent ... 

Similarly the joint probability density that Y has the value at tu and also the value y2 at r2, and so on till yn,tn, is... 

Under the Markov assumption, the hierarchy of the joint probability densities [Eqs. (4.9)] describing the evolution of the system takes the following form ... 

In physical systems it can happen that the transition probability densities are homogeneous in time and/or in space. A stochastic process X(t) is stationary if X(t) and X(t + r) obey the same probability laws for every r this means that all joint probability densities verify time translation invariance... 

An expression for the Joint Probability Density Function for the observations [ ] may be determined by transformations from e n to s n j and from s[n to [x n. This gives the likelihood function for the AR-MNL system as ... 

Notice that the arguments for the joint probabilities are ordered such that t < t2 < tn, so the order of events should be read from the right to the left. To continue we must somehow truncate the series of higher-order joint probability densities. The simplest case (often referred to as a purely random process) is one in which the knowledge of P(y,t) suffices for the solution of the problem. In particular,... 

This relation defines w2 and tells us that the joint probability density of finding y at t and 2/2 at t2 equals the probability density of finding y at t times the probability of a transition from y to y2 in time t2 — t. ... 

The essential criteria for a good fit are that the returned parameters should be as accurate as permitted by the data and that the fitting process should be robust. The theoretical limit on the uncertainties of each of the parameters in the model is given by the Cramer Rao lower bound. The Cramer Rao bound applies to any unbiased estimator 0(y) of a parameter vector, 9, using measurements, y. The measurements are described by their joint probability density function p(y 6), which is influenced by 9. 

Consider a particle whose initial dimensionless position and energy are r0 and E0, respectively. Since this particle experiences Brownian motion in the potential well, both its position and its energy will be random at any time. Therefore, the evolution of its position and energy should be described by the conditional joint probability density w(r, E t r0, Eq, 0), defined as... 

Employing the stochastic differential equations [12] and [14], the Fokker-Planck equation for the evolution of the conditional joint probability density w(r, t r0, E0 0) has the form (6)... 

As already noted, the time scale of oscillation of the particle is much smaller than its time scale of Brownian motion. Therefore, the particle undergoes very few collisions and its energy is nearly conserved during many periods of oscillations. Consequently, the conditional joint probability density can be decomposed as... 

The conditional joint probability density can, therefore, be written as... 

For scalar continuous random variables X and Y with joint probability density f (x, y), marginals and conditionals are refined as... 

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