Moment of a probability

In the mixed-effects context, the collection of population parameters is composed of a population-typical value (generally the mean) and of a population-variability value (generally the variance-covariance matrix). The mean and variance are the first two moments of a probability distribution. They build a minimal set of hyperparameters or population characteristics for it, which is sufficient (in a statistical sense) when F is taken as normal or log-normal. 

A probability distribution function contains all the information that can be known about a probabilistic system. A full distribution function, however, is rarely accessible from experiments. Generally, experiments can measure only certain averages or moments of the distribution. The nth moment of a probability distribution function (x ) is... 

Any spatial distribution of charge pix,y,z) = p r) can be described by a multipole expansion, a series in which the first term is called the monopole, the second is the dipole, the third is the quadrupole, then the octupole, etc. The various terms are moments of the distribution in the same way that the mean and standard deviation are related to the first two moments of a probability distribution (see Chapter 1). 

Moment n Formally the moment of a probability distribution. For a random variable, X, defined on a probability space, S, for an integer, k greater than or equal to 0, the fc moment about a constant, C, M)t X—C, are given by ... 

The multipole moment of rank n is sometimes called the 2"-pole moment. The first non-zero multipole moment of a molecule is origin independent but the higher-order ones depend on the choice of origin. Quadnipole moments are difficult to measure and experimental data are scarce [17, 18 and 19]. The octopole and hexadecapole moments have been measured only for a few highly syimnetric molecules whose lower multipole moments vanish. Ab initio calculations are probably the most reliable way to obtain quadnipole and higher multipole moments [20, 21 and 22]. 

Hi) Gaussian statistics. Chandler [39] has discussed a model for fluids in which the probability P(N,v) of observing Y particles within a molecular size volume v is a Gaussian fimction of N. The moments of the probability distribution fimction are related to the n-particle correlation functions and... 

To describe single-point measurements of a random process, we use the first-order probability density function p/(/). Then p/(/) df is the probability that a measurement will return a result between / and / -I- df. We can characterize a random process by its moments. The nth moment is the ensemble average of /", denoted (/"). For example, the mean is given by the first moment of the probability density function. 

Figure 5. Molecular dynamics simulation of the decay forward and backward in time of the fluctuation of the first energy moment of a Lennard-Jones fluid (the central curve is the average moment, the enveloping curves are estimated standard error, and the lines are best fits). The starting positions of the adiabatic trajectories are obtained from Monte Carlo sampling of the static probability distribution, Eq. (246). The density is 0.80, the temperature is Tq — 2, and the initial imposed thermal gradient is pj — 0.02. (From Ref. 2.)...

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... 

Thus (A2k) is a variance the 2nd central moment of the probability distribution to find the atom. 

The cumulants [26] are simple functions of the moments of the probability distribution of 5V-.C2 = (V- V))2),C3 = (V- V)f),C4 = ((]/-(]/))4) 3C22,etc. Truncation of the expansion at order two corresponds to a linear-response approximation (see later), and is equivalent to assuming V is Gaussian (with zero moments and cumulants beyond order two). To this order, the mean and width of the distribution determine the free energy to higher orders, the detailed shape of the distribution contributes. 

Statistical properties of a data set can be preserved only if the statistical distribution of the data is assumed. PCA assumes the multivariate data are described by a Gaussian distribution, and then PCA is calculated considering only the second moment of the probability distribution of the data (covariance matrix). Indeed, for normally distributed data the covariance matrix (XTX) completely describes the data, once they are zero-centered. From a geometric point of view, any covariance matrix, since it is a symmetric matrix, is associated with a hyper-ellipsoid in N dimensional space. PCA corresponds to a coordinate rotation from the natural sensor space axis to a novel axis basis formed by the principal... 

This equation can be derived from Equation IX-4 by including in the expression for the polarization the contribution due to preferential orientation of the permanent dipole moments mo in the field direction. The component of the dipole moment of a molecule in the field direction is mo cos 0, where 9 is the polar angle between the dipole-moment vector and the field direction, and the energy of interaction is —noE cos 0. The relative probability of orientation in volume element sin Qd8d (in polar coordinates) is given by the Boltzmann principle as sin 9d9dtj>. The average value of the component is hence... 

Thus the variance is always greater than that of the pure Poisson distribution with the same average. Also express the probability generating function of pn in the characteristic function of 0(a) and conclude that the moments of a are equal to the factorial moments of n compare (1.2.15). 

The quantities 7r, t, considered so far are the first moments of the probability distribution /(t) of the first-passage time. Specifically, for a one-step process there are two distributions fRttn(t) and /L>m(t) for the probabilities to arrive at R and L at a time t after starting out at site m. We derive an equation for them. By a similar argument as used above one obtains... 

This is the method of compounding moments, which avoids the explicit use of a probability in function space. It is, of course, possible to write similar equations for higher moments. 

Equations (8.10)—(8.12), tensorial ranks and boundary conditions (8.14)-(8.15) notwithstanding, embody a structure similar in format and symbolism to their counterparts for the transport of passive scalars, e.g., the material transport of the scalar probability density P (Brenner, 1980b Brenner and Adler, 1982), at least in the absence of convective transport. As such, by analogy to the case of nonconvective material transport, the effective kinematic viscosity viJkl of the suspension may be obtained by matching the total spatial moments of the probability density Pu to those of an equivalent coarse-grained dyadic probability density P j, valid on the suspension scale, using a scheme (Brenner and Adler, 1982) identical in conception to that used to determine the effective diffusivity for material transport at the Darcy scale from the analogous scalar material probability density P. In particular, the second-order total moment M(2) (sM, ) of the probability density P, defined as... 

One reason for this at first sight unexpected result is the fact that probably 70... 90% of the solute/solvent interaction term is caused by London dispersion forces, which are more or less equal for the cis and trans isomers. Another important reason is that one has to take into account higher electric moments the trans isomer has a quadrupole moment, and the cis isomer also has moments of a higher order than two. Calculations of solute/solvent interactions of both diastereomers using a reaction field model led to the conclusion that the quadrupolar contribution of the trans isomer is comparable to the dipolar contribution of the cis isomer. It has been pointed out that the neglect of solute/solvent interactions implying higher electric moments than the dipole moment can lead to completely false conclusions [202],... 

Assume for a moment that the hidden state is fixed, i.e. jt = j. Then, the evolution of a probability density p t,Y j) under the d3mamics given by (11) can be obtained as the solution of the corresponding Fokker Planck equation ... 

As already discussed below Eq, (7,5), Eq. (8.113) describes a drift diffusion process For a symmetric walk, kr = ki, v = 0 and (8.113) becomes the diffusion equation with the diffusion coefficient D = Ax"(kr + ki)/ = Ax fix. Here r is the hopping time defined from r = (k + ki). When ki the parameter v is nonzero and represents the drift velocity that is induced in the system when an external force creates a flow asymmetry in the system. More insight into this process can be obtained from the first and second moment of the probability distribution Plx, Z) as was done in Eqs (7.16)—(7.23). 

Relaxation phenomena arise already during the moment of a film drawing on a substrate when a shift pressure appears and possibilities for the cohesion break of an interface between the film and the substrate are created. During the drying, the coating should relax, forming a thin and smooth layer, thus the module of elasticity increases. Thus, the mechanical relaxation promotes decrease of the probability of fragile destruction and exfoliation of the film from the substrate [7],... 

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