Probability density distributions

The wave paeket motion of the CH eliromophore is represented by simultaneous snapshots of two-dimensional representations of the time-dependent probability density distribution... 

Heterogeneity, nonuniformity and anisotropy are based on the probability density distribution of permeability of random macroscopic elemental volumes selected from the medium, where the permeability is expressed by the one-dimensional form of Darcy s law. 

As different sources are considered, the statistical properties of the emitted field changes. A random variable x is usually characterized by its probability density distribution function, P x). This function allows for the definition of the various statistical moments such as the average. 

A laser consists in a medium where stimulated emission dominates over spontaneous emission placed inside an optical cavity which recycles the optical field. Above threshold, the photon number probability density distribution is poissonian, that means that the photon arrival time are a random variable. The probability of obtaining m photons during a given time interval is thus... 

A Fock state is a state containing a fixed number of photons, N. These states are very hard to produce experimentally for A > 2. Their photon number probability density distribution P (m) is zero everywhere except for m = N, their variance is equal to zero since the intensity is perfectly determined. Finally, the field autocorrelation function is constant... 

An analysis of the hydration structure of water molecules in the major and minor grooves in B-DNA has shown that there is a filament of water molecules connecting both the inter and the intra phosphate groups of the two strands of B-DNA. However, such a connectivity is absent in the case of Z-DNA confirming earlier MC simulation results. The probability density distributions of the counterions around DNA shows deep penetration of the counterions in Z-DNA compared to B-DNA. Further, these distributions suggest very limited mobility for the counterions and show well defined counter-ion pattern as originally suggested in the MC study. 

Quantum mechanics allows the determination of the probability of finding an electron in an infinitesimal volume surrounding any particular point in space (x,j,z) that is, the probability density at this point. Since we can assign a probability density to any point in space, the probability density defines a scalar field, which is known as the probability density distribution. When the probability density distribution is multiplied by the total number of electrons in the molecule,... 

The probability density distribution fx(x) of the random variable A- is a beta distribution with parameters m and n. Therefore... 

We now express the probability density distribution of the random variable 8lsO as... 

Figure 3.1 Energy levels and wave functions of harmonic oscillator. Heavy line bounding potential (3.2). Light solid lines quantum-mechanic probability density distributions for various quantum vibrational numbers see section 1.16.1). Dashed lines classical probability distribution maximum classical probability is observed in the zone of inversion of motion where velocity is zero. From McMillan (1985). Reprinted with permission of The Mineralogical Society of America.

LDA is the first classification technique introduced into multivariate analysis by Fisher (1936). It is a probabilistic parametric technique, that is, it is based on the estimation of multivariate probability density fimc-tions, which are entirely described by a minimum number of parameters means, variances, and covariances, like in the case of the well-knovm univariate normal distribution. LDA is based on the hypotheses that the probability density distributions are multivariate normal and that the dispersion is the same for all the categories. This means that the variance-covariance matrix is the same for all of the categories, while the centroids are different (different location). In the case of two variables, the probability density fimction is bell-shaped and its elliptic section lines correspond to equal probability density values and to the same Mahala-nobis distance from the centroid (see Fig. 2.15A). 

Quadratic discriminant analysis (QDA) is a probabilistic parametric classification technique which represents an evolution of EDA for nonlinear class separations. Also QDA, like EDA, is based on the hypothesis that the probability density distributions are multivariate normal but, in this case, the dispersion is not the same for all of the categories. It follows that the categories differ for the position of their centroid and also for the variance-covariance matrix (different location and dispersion), as it is represented in Fig. 2.16A. Consequently, the ellipses of different categories differ not only for their position in the plane but also for eccentricity and axis orientation (Geisser, 1964). By coimecting the intersection points of each couple of corresponding ellipses (at the same Mahalanobis distance from the respective centroids), a parabolic delimiter is identified (see Fig. 2.16B). The name quadratic discriminant analysis is derived from this feature. 

Probability density distribution function for bead positions in the spring-bead molecular models. 

We performed a series of theoretical studies on pump-probe diffraction patterns with a twofold objective the first aim is to evaluate the effect of electronic and vibrational excitation on electron diffraction patterns, compared to that of structural rearrangements that are the primary goal for observation in structural dynamics measurements. Secondly, we wish to explore to what extent electronic and vibrational probability density distributions are observable using the pump-probe electron diffraction methodology. Previously we have discussed the effect of electronic excitation in atomic systems,[3] and the observability of vibrational excitation in diatomic and triatomic systems.[4,5] We have now extended this work to the 8-atomic molecule s-tetrazine (C2H2N4). 

Nn = numbers of emitters per unit volume of the light source P(v) = probability density distribution function of emission per unit time and per unit frequency h = Planck s constant , = spherical coordinates... 

Fig, 7. Probability density distributions as a function ol radial distance from the nucleus for several stales of a hydrogen mom. The dashed lines are proportional in Ihe probability of finding Ihe electron in an incremental volume Jr ai ihc indicated radial distance. The solid lines are proportional to the probability for finding the electron in an incremental shell of votume 4nr2dr at the indicated radius... 

Fig. 9. Radial probability density distribution, derived from quantum-mechanical predictions. Ibr rubidium, along with Bohr planetary model for the same atom...

Fig. 3. The lattice-matched double heterostructure, where the waves shown in the conduction band and the valence band are wave functions, T(x), representing probability density distributions of carriers confined by the barriers. The chemical bonds, shown as short horizontal stripes at the AlAs—GaAs interfaces, match up almost perfecdy. The wave functions, sandwiched in by the 2.2 eV potential barrier of AlAs, never see the defective bonds of an external surface. When the GaAs layer is made so narrow that a single wave barely fits into the allotted space, the potential well is called a quantum well. Because of the match in the atomic spacings between GaAs and AlAs, 99.999% of the interfacial chemical bonds are saturated.

Mathematical construction of physical/ chemical processes that predict the range and probability density distribution of an exposure model outcome (e.g. predicted distribution of personal exposures within a study population)... 

Fig. 2.6. Schematic representation of a reaction treated in the Kramers approximation. The shape of the probability density distribution is assumed to have reached equilibrium (i.e., time independence) at the bottom of the reactant valley. Only the weight of P x. t) (total number of reactants) diminishes by activated diffusion across the barrier.

Fig. 2.7. Schematic representation of the Bagchi-Fleming-Oxtoby model used for barrierless reactions. As the probability density distribution P(x, t) (shown S shaped in this example for t = 0) moves toward the origin with a nonradiative sink S, it broadens due to the Brownian motion.

Gaussian probability density distribution. Thus, when a particle passes through a turbulent eddy, we have... 

The second term on the right-hand side of this equation takes an infinite value regardless of the values of p(t), and only the first term changes in response to the change in the probability density distribution function p(t). Therefore, the information entropy based on the continuous variable is defined as... 

It has been said in the natural world, the aim is to achieve the maximum value of information entropy. In this section, the relationship between a probability density distribution function and the maximum value of the information entropy is discussed. In the case of a mathematical discussion, it is easier to treat information entropy H(t) based on continuous variables H(t) rather than the information entropy based on discrete variables H X). In the following, H(t) is studied, and the probability density distribution function p(t) for the maximum value of information entropy H(t)max under three typical restriction conditions is shown. 

The standardized condition of the probability density distribution function is given as... 

The range of integration shows the given restrictive condition. Under these conditions, the form of the probability density distribution function p(t) for the maximum value of information entropy is investigated. By using calculus... 

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