Moments, continuous/discrete distribution

The MWD may be related mathematically to the so-called moments of a continuous or discrete distribution. If u is a random variable and F u) is its distribution function then the ith-order moment may be defined by the relation ... 

A continuous PALS spectral component is constructed as a continuous sum of discrete components and is characterized by three parameters the intensity and the first two moments of the distribution of lifetimes, that is, the centroid (mean lifetime) and the second moment (standard deviation from the mean lifetime). 

The lattice Boltzmann equation (LBE) is obtained as a dramatic simplification of the Boltzmann equation, (20.1) along with its associated equations, (20.2-20.4). In particular, it was discovered that the roots of the Gauss-Hermite quadrature used to exactly and numerically represent the moment integrals in (20.4), corresponds to a particular set of few discrete particle velocity directions c in the LBE [11]. Eurther-more, the continuous equihbrium distribution (20.3) is expanded in terms of the fluid velocity m as a polynomial, where the continuous particle velocity is replaced by the discrete particle velocity set ga obtained as discussed above, i.e., becomes f 1. After replacing the continuous distribution function / =f x, t) in (20.1) by the discrete distribution function/ =f x, (20.1) is integrated by considering... 

Second moments such as the variance are important for understanding heat capacities (Chapter 12), random walks (Chapters 4 and 18), diffusion (Chapter 18), and polymer chain conformations (Chapters 31-33). Moments higher than the second describe asymmetries in the shape of the distribution. Examples 1.20, 1.21, and 1.22 show calculations of means and variances for discrete and continuous probability distributions. 

Let us look first at the transition of the original definitions as integrals over the charge density, Eqs. (4.5), (4.6) and (4.8), to quantum mechanics that we will illustrate for the example of the electric dipole moment. In the Born-Oppenheimer approximation, Section 2.2, the electrons in a molecule form a continuous charge distribution whereas the discrete nuclear charges are located at fixed points Rk- The expression, Eq. (4.5) for the a-component of the electric dipole moment can therefore be rewritten as... 

Moments 92. Common Probability Distributions for Continuous Random Variables 94. Probability Distributions for Discrete Random Variables. Univariate Analysis 102. Confidence Intervals 103. Correlation 105. Regression 106. 

Another kind of situation arises when it is necessary to take into account the long-range effects. Here, as a rule, attempts to obtain analytical results have not met with success. Unlike the case of the ideal model the equations for statistical moments of distribution of polymers for size and composition as well as for the fractions of the fragments of macromolecules turn out normally to be unclosed. Consequently, to determine the above statistical characteristics, the necessity arises for a numerical solution to the material balance equations for the concentration of molecules with a fixed number of monomeric units and reactive centers. The difficulties in solving the infinite set of ordinary differential equations emerging here can be obviated by switching from discrete variables, characterizing macromolecule size and composition, to continuous ones. In this case the mathematical problem may be reduced to the solution of one or several partial differential equations. 

Continuous distribution functions Some experiments, such as liquid chromatography or mass spectrometry, allow for the determination of continuous or quasi-continuous distribution functions, which are readily obtained by a transition from the discrete property variable X to the continuous variable X and the replacement of the discrete statistical weights g, by the continuous probability density g(X). For simplicity, we assume g(X) as being normalized J ° g(X)dX = 1. Averages and moments of a quantity Y(X) are defined by analogy to the discrete case as... 

It seems that there is a need to reexamine, some of the basic quantities used in transport processes, like Thiele numbers, attempting to connect them to more chemical quantities. For example, the macroscopic quantity, e the dielectric constant, can be interpreted in terms of dipole moment distribution, and the dipole moment has immediate structural implications. Now to talk of a dielectric constant in the interaction of two atoms would be a rather useless exercise, since the dilectric constant is a continuous matter concept, not a discrete matter concept. In the same... 

The reconstructed distribution function may be continuous (EQMOM) or discrete (QMOM), but we will assume that it is always realizable (i.e. nonnegative). For the case in which / is a set of weighted delta functions, the computation of the moments and is trivial. With EQMOM the integrals are evaluated using... 

The spinodal curve and the critical points (including multiple critical points) only depend on few moments of the molar-mass distribution of the polydisperse system while the cloud-point curve the shadow curve and the coexistence curves are strongly influenced by the whole curvature of the distribution function. The methods used that include the real molar-mass distribution or an assumed analytical distribution replaced by several hundred discrete components have been reviewed by Kamide. In the 1980s continuous thermodynamics was applied, for example, by Ratzsch and Kehlen to calculate the phase equilibrium of a solution of polyethene in supercritical ethene as a function of pressures at T= 403.15 K. The Flory s model was used with an equation of state to describe the poly-dispersity of polyethene with a a Wesslau distribution. Ratzsch and Wohlfarth applied continuous thermodynamics to the high-pressure phase equilibrium of ethene [ethylene]-I-poly(but-3-enoic acid ethene) [poly(ethylene-co-vinylace-tate)] and to the corresponding quasiternary system including ethenyl ethanoate [vinylacetate]. In addition to Flory s equation of state Ratzsch and Wohlfarth also tested the Schotte model as a suitable means to describe the phase equilibrium neglecting the polydispersity with respect to chemical composition of the... 

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