Joint distribution functions, example

Another instructive example concerns the joint distribution function of the pair of time functions Zx(t) and Z2(t) defined by... 

The Poisson process represents only one possible way of assigning joint distribution functions to the increments of counting functions however, in many problems, one can argue that the Poisson process is the most reasonable choice that can be made. For example, let us consider the stream of electrons flowing from cathode to plate in a vacuum tube, and let us further assume that the plate current is low enough so that the electrons do not interact with one another in the... 

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... 

This is an example of a joint or sirmiltaneous distribution function (Cramer, 1955). It is not necessary to include ni as one of the independent variables because of the relationship between u and ... 

Conditional and Joint Pmbability Distributions In addition to its graphical structure, a Bayesian network needs to be speerfied by the conditional probability distribution of eaeh node given its parents. Let A and D be variables of interest with a direct causal (parental) relationship in Example 11.6. This relationship can be represented by a conditional probabUity distribution P D A) which represents the probabilistic distribution of child node D given the information of parent node A. When both child and parent nodes arc discrete variables, a contingeney table can summarize the conditional probabdities for aU possible states given each of its parent node states. For continuous variables, a eonditional probability density function needs to be defined. For the combination of continuous and discrete nodes, a mixture distribution, for example, mixture normal distribution, will be required (Imoto et al., 2002). 

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. 

Suppose that we have a postulated model p y 6) for predicting the probability distribution of future observation sets y for each permitted value 6 of a list of parameters. (An illustrative model is given in Example 5.1.) Suppose that a prior probability function p 9) is available to describe our information, beliefs, or ignorance regarding 6 before any data are seen. Then, using Eq. (5.1-7), we can predict the joint probability of y and 6 either as... 

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... 

The concentration-distance curves method is based on the measurement of the distribution of the diffusant concentration as a function of time. Light interference methods, as well as radiation adsorption or simply gravimetric methods, can be used for concentration measurements. Various sample geometries can be used, for example semiinflnite solid, two joint cylinders with the same or different material, and so on. The analysis is based on the solution of Pick s equation. 

The organization distributes distinct functions across its network of partners, suppliers, joint contractors, subcontractors or distributors. This is the case at Airbus Toulouse (France), which divides tasks between Airbus and its network of suppliers. Production processes are distributed across the manufacturer s internal sites. Airbus is the prime example of the lead network firm it delegates its non-strategic sub-functions in order to focus on its core activities. 

The most appropriate interpretation of biocompatibility for PE biomaterial applications is that the biocompatibility be defined in terms of the success of a device in fulfilling its intended function. For example, for a hip joint prosthesis, one must take into consideration the fatigue resistance of the device, its corrosion resistance, the distribution of the stresses transferred to the bone by the device, the solid angle of mobility provided, and the overall success of the device in restoring a patient to an ambulatory state. The performance of a hip joint prosthesis might also be assessed in terms of the tissue reaction to acetabular cup. The performance of individual materials is sometimes referred to as biocompatibility and sometimes as bioreaction . Hardness, shape, porosity, and specific implant site are very important [58]. 

New pipe concepts with multilayer structures, or so-called functional layers, have helped to expand the applications of HOPE. Pipes with special protective layers (on the outside and/or inside) can be laid without a sand bed under some circumstances. This significantly reduces the pipe laying costs. There have also been new developments in joining technology. For example, electric welding fittings up to 700 mm are now available. Electrofusion jointing of PE pipes for gas distribution is also applied. 

We will use the two-parameter case to show what happens when there are multiple parameters. The inference universe has at least four dimensions, so we cannot graph the surface on it. The likelihood function is still found by cutting through the surface with a hyperplane parallel to the parameter space passing through the observed values. The likelihood function will be defined on the the two parameter dimensions as the observations are fixed at the observed values and do not vary. We show the bivariate likelihood function in 3D perspective in Figure 1.8. In this example, we have the likelihood function where 9 is the mean and 62 is the variance for a random sample from a normal distribution. We will also use this same curve to illustrate the Bayesian posterior since it would be the joint posterior if we use independent flat priors for the two parameters. 

The uncertainty of a parameter can be characterised by the upper and lower limits of the parameter or by the expected value and the variance of the parameter. Such descriptions of individual parameter uncertainty can, for example, be obtained from the data evaluation sources introduced in Chap. 3. The joint probability density function (pdf) of parameters gives the most complete information about the uncertainty of a parameter set. Methods of uncertainty analysis provide information about the uncertainty of the results of a model knowing the uncertainty of its input parameters. If such a lack of knowledge of model inputs is propagated through the model system then a model output becomes a distribution rather than a single value. Measures such as output variance can then be used to represent output uncertainty. 

---