Conditional probability density

Ponderomotive force, 382 Position operator, 492 in Dirac representation, 537 in Foldy-Wouthuysen representation, 537 spectrum of, 492 Power, average, 100 Power density spectrum, 183 Prather, J. L., 768 Predictability, 100 Pressure tensor, 21 Probabilities addition of, 267 conditional, 267 Probability, 106... 

Wall-boundary conditions in probability density function methods and application to a turbulent channel flow. Physics of Fluids 11, 2632-2644. 

One of the most important concepts of any probability theory is the conditional probability. In the density-based approach we can introduce the conditional density. If densities p D) and p(H) (24) exist, p(H) 0 and the following limit p D IH) exists, then we call it conditional density ... 

For polyhedra the situation is similar to usual probability theory densities p(D) and p(H) always exist and if p H) / 0 then conditional density exists too. For general measurable sets the situation is not so simple, and existence of p(D) and p(H)= 0 does not guarantee existence of p(D IH). 

The first difficulty derives from the fact that given any values of the macroscopic expected values (restricted only by broad moment inequality conditions), a probability density always exists (mathematically) giving rise to these expected values. This means that as far as the mathematical framework of dynamics and probability goes, the macroscopic variables could have values violating the laws of phenomenological physics (e.g., the equation of state, Newton s law of heat conduction, Stokes law of viscosity, etc.). In other words, there is a macroscopic dependence of macroscopic variables which reflects nothing in the microscopic model. Clearly, there must exist a principle whereby nature restricts the class of probability density functions, SF, so as to ensure the observed phenomenological dependences. 

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... 

Many trajectories are necessary to describe all the different events that are summed up to form a unique wave describing the global chemical reaction under observable conditions in quantum mechanics. In this respect, a set of classical trajectories which spread around a mean trajectory in classical mechanics corresponds roughly to the quantum mechanical spreading (through space or time) of the density probability function around its center. 

Often the probability of an event depends on one or more related events or conditions. Such probabilities are called conditional We will write p A B) for the probability of event A given B (or the probability density of A given B if the sample space of A is continuous). 

P(d/m) is a conditional density of probability for the data d, given the model m. It means that it is the probability density of theoretical data d to be expected from a given model m. 

P(m/d) is a conditional density of probability for a model m, given the data d. According to the Bayes theorem, the following equation holds ... 

The so presented ELF is mainly based on an interpretation of the conditional pair probability density for electrons of the same spin. A conceptually different interpretation was put forward by Savin et al.23 who realized that the term Da could be generalized for any density p independent of the spin as... 

Figure 11. Distance required to remove 99% of particles from suspension (L ) as a function of suspended particle radius for two chemical conditions (attachment probabilities). Flow rate = O.lm/day, media radius = 0.025 cm, temperature = 25 °C, particle density = 1.05 g/cm3, and aquifer porosity = 0.4. (Reproduced with permission from reference 29. Copyright 1987.)...

The conditional quadrature method of moments (CQMOM) is based on the concept of a conditional density function (Yuan Fox, 2011). Conditional density functions represent, in turn, the probability of having one internal coordinate within an infinitesimal limit when one or more of the other internal coordinates are fixed and equal to specific values. For example, in the case of a generic NDF the expression... 

The collimation is characterized by the function (fc, 0, 0 ) which is the density probability that an incident particle with wave number k, deviated in the sample by an angle 0, encounters the detector whose direction is given by 0. In such conditions, the number of scattered particles that encounter the sample in a time interval t is... 

In this context, when the inverse of difference in local kinetic terms is involved, the ELF is interpreted as the error in localization of electrons within traps rather than where they have peaks of spatial density, as is frequently misinterpreted in literature (Santos et al., 2000 Scemama et al., 2004 Soncini Lazzeretti, 2003 Silvi, 2003), albeit recent extensions of ELF have used the correlated (HF) wave functions, through the conditional pair probability, however not using the kinetic energy approach (Matito et al, 2006 Kohout et al., 2004 Jensen, 2005). 

The potential for reduced size and complexity is important if BN models are to be applied on a grand scale, modelling the complete set of loss events faced by a financial institution. Because each node in the network needs to be assigned a (conditional) density or probability mass function a simple structure may provide more efficient modelling as the required input is reduced. Hence the model suggested in this paper may potentially reduce the effort needed in constructing the BN model. 

The optimal decision rule is found by the use of the Bayes approach. The unknown parameter, whose value we want to decide about, is random variable Y e 0,1 with probability function q(y). The decision will be made on the basis of the value of random vector X with the density function r(x). Let r(x y) be the conditional density function of X on condition Y =y, S JiF 0,1 the decision function andi/ the set of all decision functions 5 0,1. The loss... 

To quantify flood risks, state of the art modeling techniques combine (i) probability density functions of hydraulic conditions (ii) probability density functions of the variables that determine the load bearing capacity of a flood defense, (iii) fault tree models to analyze failure modes, and (iv) flood propagation models, land-use data and loss functions to relate flood characteristics and land-use data to the consequences of flood scenarios (e.g. Van Manen Brinkhuis 2005). This paper focuses on the quantification of loss of life for a given flood scenario. The quantification of flood probabilities and flood characteristics (such as flow velocities, rise rates, and inundation depths) is outside the scope of the present paper and discussed in e.g. Steenbergen et al. (2004). 

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