Probability density moments

The next example will illustrate the technique of calculating moments when the probability density function contains Dirac delta functions. The mean of the Poisson distribution, Eq. (3-29), is given by... 

The physical interpretation of these joint moments is similar in every respect to the interpretation already given for moments of the form ak = E[k]. Thus, a . .. provides a measure of the center of mass of the joint probability density function p 1,...,second order central moments provide a measure of the spread of this density function about its center of mass.30... 

Although we cannot easily obtain expressions for the probability density functions of Y(t), it is a simple matter to calculate its various moments. We shall illustrate this technique by calculating all possible first and second moments of Y(t) i.e., E[7(t)] and E[Y(t)Y(t + r)], — oo < v < oo. The pertinent characteristic function for this task is MYQt (hereafter abbreviated MYt) given by... 

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. 

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. 

The steady-state probability distribution for a system with an imposed temperature gradient, pss(r p0, pj), is now given. This is the microstate probability density for the phase space of the subsystem. Here the reservoirs enter by the zeroth, (10 = 1 /k To, and the first, (i, = /k T, temperatures. The zeroth energy moment is the ordinary Hamiltonian,... 

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... 

A continuous Markov process (also known as a diffusive process) is characterized by the fact that during any small period of time At some small (of the order of %/At) variation of state takes place. The process x(t) is called a Markov process if for any ordered n moments of time t < < t < conditional probability density depends only on the last fixed value ... 

The transition probability density becomes Dirac delta function for coinciding moments of time (physically this means small variation of the state during small period of time) ... 

Moments of the first passage time may be expressed from the probability density wt(U xo) as... 

Finally, for additional support of the correctness and practical usefulness of the above-presented definition of moments of transition time, we would like to mention the duality of MTT and MFPT. If one considers the symmetric potential, such that (—oo) = <1>( I oo) = +oo, and obtains moments of transition time over the point of symmetry, one will see that they absolutely coincide with the corresponding moments of the first passage time if the absorbing boundary is located at the point of symmetry as well (this is what we call the principle of conformity [70]). Therefore, it follows that the probability density (5.2) coincides with the probability density of the first passage time wT(f,xo) w-/(t,xo), but one can easily ensure that it is so, solving the FPE numerically. The proof of the principle of conformity is given in the appendix. 

The cumulants [2,43] of decay time sen are much more useful for our purpose to construct the probability P(t. xq)—that is, the integral transformation of the introduced probability density of decay time wT(t,xo) (5.2). Unlike the representation via moments, the Fourier transformation of the probability density (5.2)—the characteristic function—decomposed into the set of cumulants may be inversely transformed into the probability density. 

Probably, a similar procedure was previously used (see Refs. 1 and 93-95) for summation of the set of moments of the first passage time, when exponential distribution of the first passage time probability density was demonstrated for the case of a high potential barrier in comparison with noise intensity. 

We consider the process of Brownian diffusion in a potential cp(x). The probability density of a Brownian particle is governed by the FPE (5.72) with delta-function initial condition. The moments of transition time are given by (5.1). 

Thus, we have proved the principle of conformity for both probability densities and moments of the transition time of symmetrical potential profile and FPT of the absorbing boundary located at the point of symmetry. 

In a turbulent flow, the local value (i.e., at a point in space) of the mixture fraction will behave as a random variable. If we denote the probability density function (PDF) of by f - Q where 0 < ( < 1, the integer moments of the mixture fraction can be found by integration ... 

In order to compare various reacting-flow models, it is necessary to present them all in the same conceptual framework. In this book, a statistical approach based on the one-point, one-time joint probability density function (PDF) has been chosen as the common theoretical framework. A similar approach can be taken to describe turbulent flows (Pope 2000). This choice was made due to the fact that nearly all CFD models currently in use for turbulent reacting flows can be expressed in terms of quantities derived from a joint PDF (e.g., low-order moments, conditional moments, conditional PDF, etc.). Ample introductory material on PDF methods is provided for readers unfamiliar with the subject area. Additional discussion on the application of PDF methods in turbulence can be found in Pope (2000). Some previous exposure to engineering statistics or elementary probability theory should suffice for understanding most of the material presented in this book. 

To account for the effect of a sufficiently broad, statistical distribution of heterogeneities on the overall transport, we can consider a probabilistic approach that will generate a probability density function in space (5) and time (t), /(i, t), describing key features of the transport. The effects of multiscale heterogeneities on contaminant transport patterns are significant, and consideration only of the mean transport behavior, such as the spatial moments of the concentration distribution, is not sufficient. The continuous time random walk (CTRW) approach is a physically based method that has been advanced recently as an effective means to quantify contaminant transport. The interested reader is referred to a detailed review of this approach (Berkowitz et al. 2006). 

When characterising probability density functions, it is frequently more convenient to describe a function in terms of its moments about the mean Ml, oi centralised moments, rather than in terms of moments about the origin. These are defined by... 

The alternative is the use of a descriptive mathematical model without any relation with the solution of the transport equation. On the analog of the characterization of statistical probability density functions a peak shape f(t) can be characterized by moments, defined by ... 

Analytical solutions of quantum Fokker-Planck equations such as Eq. (63) are known only in special cases. Thus, some special methods have been developed to obtain approximate solutions. One of them is the statistical moment method, based on the fact that the equation for the probability density generates an infinite hierarchic set of equations for the statistical moments and vice versa. 

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. 

The question of determinateness presents itself as follows Let the initial (t = 0) values of all the macroscopically independent macroscopic variables be given the equations of macroscopic physics (thermal and hydro-dynamic equations, etc.) show that these variables evolve deterministically with t 0. Yet there are infinitely many different probability densities ( ")t = 0 which have the moments, etc., coinciding with the set of given initial macroscopic values. Each evolves (by Liouville s equation) differently, and hence may induce a different set of macroscopic expected values at / > 0. By what principle of natural selection is the class of probability densities so restricted as to restore macroscopic determinacy ... 

This I(S) is always defined when there is at least one distribution for which. S is true but it need not be finite. Thus, if the domain of definition of. S is the set of probability densities p(x) on the whole x axis, a trivial 5 (i.e., true for every p(x)) and also an S which merely gives the value of the first moment, has I(S) = — oo. On the other hand, if S states that the second moment is close to zero, /( S ) is very large, and I(S) -> + oo as this moment approaches zero. Of course there is no p(x) having a zero second moment (only a point distribution, which is not a p(x)). Thus it might seem natural to define I(S) as + oo when S defines an empty set. Then every S without exception has a unique I(S). 

Here, is the mean collision frequency of molecules of the type A with B vc(u) is the collision frequency of molecule A with A s initial speed being v P(v) is the probability density for AB collisions with A s initial speed being v K(v) is the intracollisional spectrum at zero frequency, per collision, with initial speed v A(v) is the mean projection of the integrated dipole moment induced by collisions with initial speeds v on the velocity... 

Moment Generating Function. For the random variable X. with probability density function/(x), if the function M(t) = E[ea] exists, it is the moment generating function. Assuming the function exists, it can be shown that d M(l)/df t=0 = E x. Find the moment generating functions for... 

The quantity pv C is the unpaired w-electron spin density at the carbon atom to which the hydrogen atom in question is bonded p c is defined as 1 times the fractional number of unpaired it electrons on the carbon atom, with the sign being determined by whether the net unpaired spin at the carbon atom is in the same or opposite direction as the spin vector of the molecule. The term -electron spin density is somewhat misleading in that pn is not an electron probability density (which is measured in electrons/cm3), but rather is a pure number. The semiempirical constant Q (no connection with nuclear quadrupole moments) is approximately —23 G. 

Remark. Mathematically the probability density in function space and the integration over all functions is not defined. The reason is that this inadvertently introduces an overwhelming amount of very rapidly varying functions u(r). Physically they are meaningless, because (1.3) defines u(r) as an interpolation of numbers on a grid. The problem is therefore to find a mathematically consistent and physically satisfactory method for restricting function space to sufficiently smooth functions. This problem need not be solved, however, because the resulting equations for the moments lead to correct results. 

The above-described pair problem is treated by the Smoluchowski equation [3, 19] - see Fig. 1.10. It operates with the probability densities (Fig. 1.11) and contains the recombination rate characterizing particle motion. Knowledge of the probability density to find a particle at a given point at time moment t gives us (by means of a trivial integration over reaction volume) the quantity of our primary interest - survival probability of a particle in the system with... 

To describe the geminate kinetics, let us start with the motion equations for the probability density W(fX, rB, i), where W(fX, rB, )dfXdrB gives the probability to find at the moment t particle A at volume dfX centred at fX and particle B at drB centred at fie. The relevant motion equation reads [64]... 

The concept of probability density may also be utilized to describe the discrete spectrum or, particularly, to solve the inverse problem of the approximate reconstruction of the spectrum from its envelope line using a certain number of its lower moments. The density function of the energy levels of the discrete spectrum may have the form... 

The essential difference between the two transition probability densities lies in the fact that for the gaussian distribution pw r, ) the different moments E[Xm], m = 1, 2,. . . , n, exist, while for the Cauchy distribution pc(j, x) they do not exist. The Levy distributions characterized by p(t, k) = exp -a k qT) with 0< <2U 127 128 play a prominent role in the theory of relaxation processes.129 133... 

In the Bom interpretation (Section 4.2.6) the square of a one-electron wavefunction ij/ at any point X is the probability density (with units of volume-1) for the wavefunction at that point, and j/ 2dxdydz is the probability (a pure number) at any moment of finding the electron in an infinitesimal volume dxdydz around the point (the probability of finding the electron at a mathematical point is zero). For a multielectron wavefunction T the relationship between the wavefunction T and the electron density p is more complicated, being the number of electrons in the molecule times the sum over all their spins of the integral of the square of the molecular wavefunction integrated over the coordinates of all but one of the electrons (Section 5.5.4.5, AIM discussion). It can be shown [9] that p(x, y, z) is related to the component one-electron spatial wavefunctions ij/t (the molecular orbitals) of a single-determinant wavefunction T (recall from Section 5.2.3.1 that the Hartree-Fock T can be approximated as a Slater determinant of spin orbitals i/qoc and i// /i) by... 

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