Probability density function interpretation

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... 

The knowledge required to implement Bayes formula is daunting in that a priori as well as class conditional probabilities must be known. Some reduction in requirements can be accomplished by using joint probability distributions in place of the a priori and class conditional probabilities. Even with this simplification, few interpretation problems are so well posed that the information needed is available. It is possible to employ the Bayesian approach by estimating the unknown probabilities and probability density functions from exemplar patterns that are believed to be representative of the problem under investigation. This approach, however, implies supervised learning where the correct class label for each exemplar is known. The ability to perform data interpretation is determined by the quality of the estimates of the underlying probability distributions. 

A time-independent wave function is a function of the position in space (r = x,y,z) and the spin degree of freedom, which can be either up or down. The physical interpretation of the wave function is due to Max Born (25, 26), who was the first to interpret the square of its magnitude, > /(r)p, as a probability density function, or probability distribution function. This probability distribution specifies the probability of finding the particle (here, the electron) at any chosen location in space (r) in an infinitesimal volume dV= dx dy dz around r. I lu probability of finding the electron at r is given by )/(r) Id V7, which is required to integrate to unity over all space (normalization condition). A many-electron system, such as a molecule, is described by a many-electron wave function lF(r, r, l .I -.-), which when squared gives the probability den-... 

The new model has also been applied to the calculation of thermally averaged probability density functions for the out-of-plane inversion motion of the CH and H3O ions [9]. Such probability densities can be obtained experimentally by means of Coulomb Explosion Imaging (CEI) techniques (see, for example, Refs. [10,11]), and the results in Ref. [9] will be useful in the interpretation of the resulting images, just as analogous calculations of the bending probability distribution for the CHj ion were instrumental in the interpretation of its CEI images (see Refs. [9,12] and references therein). 

The probability of finding the particle in an infinitesimal cube of sides dx, dy, dz is ij/2dxdydz, and the probability of finding the particle somewhere in a volume V is the integral over that volume of iJ/2 with respect to dx, dy, dz (a triple integral) ij/2 is thus a probability density function, with units of probability per unit volume. Bom s interpretation was in terms of the probability of a particular state, Pauli s the chemist s usual view, that of a particular location. 

The interpretation of any distribution function as a probability density function in phase space leads to the requirement... 

The matrices P, Q, and R can be interpreted as covariance matrices of corresponding random variables, for which probability density functions are normal subject to truncation of their tail ends, dictated by the constraints of Eq. (168). 

Pore Size Distribution. The pore structure is sometimes interpreted as a characteristic pore size, which is sometimes ambiguously called porosity. More generally, pore structure is characterized by a pore size distribution, characteristic of the sample of the porous medium. The pore size distribution/ ) is usually defined as the probability density function of the pore volume distribution with a corresponding characteristic pore size 6. More specifically, the pore size distribution function at 5 is the fraction of the total pore volume that has a characteristic pore size in the range of 5 and 5 + dd. Mathematically, the pore size distribution function can be expressed as... 

It should be noted that we integrate with respect to the forward variable y in (3.236). In this case, (3.236) has a very nice probabilistic interpretation. Consider the Brownian motion B t), which is a stochastic process with independent increments, such that B(t + s) - B(s) is normally distributed with zero mean and variance 2Dt. The corresponding transition probability density function p y, t x) is given by (3.237). Therefore the solution (3.236) has a probabilistic representation... 

In this equation Fn y) is the cumulative probability distribution function of the strength or P[R < y] and fg (y) is the probability density function of the load efiVet or P y < 5 < y + 6y]. The interpretation of this equation in words is the suir.mation over all y, of the probability that the strength effect is less than y and the load effect is equal to y, assuming that the two effects are independent of each other. In the steel beam example the strength effect R = fy Zp and S — M/%, the effect for this limit state being that of bending moment. For other limit states the effects may be stress, strain, deflection, vibration etc. 

After the seminal structure building of the QS formalism, several additional studies appeared over time, which developed new theoretical details. Especially noteworthy is the concept of vector semispace (VSS). This mathematical construction will be shown to be the main platform on which several QS ideas are built, related in turn, to probability distributions and hence to quantum mechanical probability density functions. Such quantum mechanical density distributions form a characteristic functional set, which can be easily connected to VSS properties. Construction of the so-called quantum objects (QO) and their collections the QO sets (QOS) (see, for example, Carbo-Dorca ), easily permit the interpretation of the nature of quantum similarity measures for relationships between such quantum mechanically originated elements. Within quantum similarity context, QOS appear as a particular kind of tagged sets, where objects are submicroscopic systems and their density functions become tags. 

Method 2 differs from method 1 in that it treats probabilities as subjective figures (a Bayesian interpretation of probability), rather than as relative frequencies. In method 2, the upper, mean, and lower values from figure 1 are assumed to belong to a single probability density function. This probability density function corresponds to a single FN-curve (no bandwidths). This time, the FN curve does not show imcertain cmnu-lative frequencies and consequences. Rather, it now represents the imcertainty (related to consequences) itself... 

The Liouville equation, Eq. (2.6), describes the behavior of the collection of phase points as they move through a multidimensional space, or phase space, representing the position and momentum coordinates of all molecules in the system. The phase points tend to be concentrated in regions of phase space where it is most likely to find the N molecules with a certain momentum and position. Thus, the density function pN can be interpreted (aside from a normalization constant) as a probability density function, i.e., pjvdr- dp is proportional to the probability of finding a phase point in a multidimensional region between (r, p ) and (r - - d r, + d p ) at any time t. 

We have already indicated that the square of the electromagnetic wave is interpreted as the probability density function for finding photons at various places in space. We now attribute an analogous meaning to for matter waves. Thus, in a one-dimensional problem (for example, a particle constrained to move on a line), the probability that the particle will be found in the interval dx around the point xi is taken to be ir x ) dx. If i/ is a complex fimction, then the absolute square, is used instead of... 

Fluidized-bed reactors. Tracers have been used to evaluate backmixing of the gas, gas-solid contacting, solids backmixing, and the contact time distribution, i.e. the probability density function of sojourn times in the reaction environment. The literature in this area is rather abundant and unfortunately to some extent confused. No attempt is made here at a comprehensive review. Rather, we want to point out some highlights and indicate that the interpretation of all reported results is straightforward in view of the theory reviewed in Section 6.1.1. 

On the other hand, false coverage probabilities do not allow any practical interpretation. In particular, an experimenter would like to know, in practical terms, how well a method performs with respect to others and how its performance is affected by the sample size. Let s take a closer look at the behavior oi Ip. K simulation of Ip values, which will be explained later, yielded the probability density functions for the two methods illustrated in Fig. 1. For this simulation the parameters are set as m = 5, CTo = 100, p = Q.Ql, and the sample size is chosen as = 20. The density function of Ip estimated by the ML method is slightly to the right of the function estimated by the GLR method 1. Therefore, for any a p significant difference from the viewpoint of an experimenter. Further, Method 1 is one of the worst methods in Table 1, so the difference between the ML method and the GLR methods with weighted least squares is expected to be smaller. One practical way for quantifying the difference is to compute the sample size necessary for the ML method to have approximately the same density function as the GLR method 1. 

The probability density functions of the observations are generally unknown, but the Gauss-Markov theorem ensures that least-squares is always an acceptable estimator. However, the results of least squares are strongly influenced by discordant observations, so-called outliers. The robust-resistant techniques use weight-modification functions of O—Cy which progressively down-weight outliers. Tnese functions implicitly define probability functions p. They may alternatively be interpreted as an appreciation of the reliability of certain measurements. This approaches the frequently used option to simply omit discordant observations because they are judged to be unreliable. 

The product ifnj/ may be interpreted as the probability density function for the electron. The normalization coefficients A, B, C were chosen so that... 

In order to interpret the square of a wavefunction as a probability density function, conditions exist on the allowed forms of wavefunctions. As mentioned earlier, every wavefunction must be smooth, continuous, and single-valued when the potential has no infinite jumps. In the case of the step potential in Figure 8.4, having these conditions hold at the boundaries between regions implies constraints on the wavefunctions, specifically on the undetermined constants in Equations 8.30 through 8.32. For the wavefunction to be continuous and single-valued the following must hold ... 

Finally, by using many rectangles and drawing a smooth curve through their tops, we obtain the particle size distribution curve that is the graphical representation of the fiequency function, or probability density function. Figure 4.4 is an accurate picture of how the particles are distributed among the various sizes it has the same characteristics as Fig. 4.3, but may be amenable to mathematical interpretation. 

In our discussion of the electron density in Chapter 5, I mentioned the density functions pi(xi) and p2(xi,X2). I have used the composite space-spin variable X to include both the spatial variables r and the spin variable s. These density functions have a probabilistic interpretation pi(xi)dridii gives the chance of finding an electron in the element dri d i of space and spin, whilst P2(X], X2) dt] d i dt2 di2 gives the chance of finding simultaneously electron 1 in dri dii and electron 2 in dr2di2- The two-electron density function gives information as to how the motion of any pair of electrons is correlated. For independent particles, these probabilities are independent and so we would expect... 

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