Symmetric probability distribution

FIGURE 8.2 Both Is and 2p orbitals give symmetric probability distributions. Combining the two into a superposition state breaks this symmetry, because the 2p orbital constructively interferes with Is in one lobe and destructively interferes in the other. 

There are several possible ways to characterize a non-symmetrical probability distribution by a single number. The three different speeds discussed above serve this purpose for the Maxwell-Boltzmann distribution. Because the distribution is non-symmetrical, they are close to each other but are not equal. They stand in the ratio ... 

In the absence of an external force, the probability of moving to a new position is a spherically symmetrical Gaussian distribution (where we have assumed that the diffusion is spatially isotropic). 

Another simple example is the traiditional two-dimensional random-walk on a four-neighbor Euclidean lattice [toff89]. Despite the fact that the underlying lattice is symmetric only with respect to rotations that are multiples of 90 deg, the probability distribution p(s, y) for a particle that begins its random walk at the origin becomes circularly symmetric in the limit as time t —> oo p x,y,t) —> (see figure 12.12). 

Fig. 12.12. A circularly symmetric Gaussian probability distribution p x,y) describing a two-dimensional random walk emerges for large times on the macroscopic level, despite the fact that the underlying Euclidean lattice is anisotropic.

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... 

As Eq. (2.31) shows, the Gram-Charlier temperature factor is a power-series expansion about the harmonic temperature factor, with real even terms, and imaginary odd terms. This is an expected result, as the even-order Hermite polynomials in the probability distribution of Eq. (2.30) are symmetric, and the odd-order polynomials are antisymmetric with respect to the center of the distribution. 

If the structure of the helium atom were exactly described by the symbol la9 and that of neon by 1 a22a 2p these atoms would have spherically symmetrical electron distributions.24 However, the mutual repulsion of the two electrons in the atom causes them to avoid one another the wave function for the atom corresponds to a larger probability for the two electrons to be on opposite sides of the nucleus than on the same side (for the same values of the distances of the two electrons from the nucleus, there is greater probability that the angle described at the nucleus by the vectors to the electrons is greater than 90° than that it is less than 90°). This effect, which is called correla-... 

It is an interesting Tact that just as the single s orbital is spherically symmetric, the summation or electron density of a set or three p orbitals, five d orbitals, or seven f orbitals is also spherical (UnsBld s theorem). Thus, although it might appear as though an atom such as neon with a filled set of sand p orbitals would have a lumpy electron cloud, the total probability distribution is perfectly spherical... 

Fig. 23.8 Probability distribution Nn x,y z) 2 for the intrashell wavefunction N = n = 6 in the x = 0 plane corresponding to the collinear arrangement rn = rj +r2. The axes have a quadratic scale to account for the wave propagation in coulombic systems, where nodal distances increase quadratically. The fundamental orbit (-----------) (as) as well as the symmetric stretch motion (------) (ss) along the Wannier ridge are overlayed on the figure (from...

Solving Equation 6.5 gives (x) = L/2, which is also obvious from inspection. All of the probability distributions for the stationary states are symmetric about the center of the box. 

Note that the average velocity for motion in both directions is zero, since the probability distribution is symmetrical with respect to the direction of the motion. 

There is a theoretical study on the asymptotic shape of probability distribution for nonautocatalytic and linearly autocatalytic systems with a specific initial condition of no chiral enantiomers [35,36]. Even though no ee amplification is expected in these cases, the probability distribution with a linear autocatalysis has symmetric double peaks at 0 = 1 when ko is far smaller than k -,kototal number of all reactive chemical species, A, R, and S. This can be explained by the single-mother scenario for the realization of homo chirality, as follows From a completely achiral state, one of the chiral molecules, say R, is produced spontaneously and randomly after an average time l/2koN. Then, the second R is produced by the autocatalytic process, whereas for the production of the first S molecule the... 

Random error — The difference between an observed value and the mean that would result from an infinite number of measurements of the same sample carried out under repeatability conditions. It is also named indeterminate error and reflects the - precision of the measurement [i]. It causes data to be scattered according to a certain probability distribution that can be symmetric or skewed around the mean value or the median of a measurement. Some of the several probability distributions are the normal (or Gaussian) distribution, logarithmic normal distribution, Cauchy (or Lorentz) distribution, and Voigt distribution. Voigt distribution is... 

Given n points in 2D whose positions are given as normal probability distributions Q At(P ,A() i = 0... n -1, we find the C -symmetric configuration of points P( q i which is most probable. Denote by CO the center of mass of P. ... 

Figure 19 shows an example of varying the probability distribution of a measurement on the resulting symmetric shape. Figure 19a shows the most probable C2-symmetric shape for the set of measurements of Figure 18a. Figures 19b-d show the most probable C2-symmetric shape after varying the distribution of the bottom measurement.

Here the question of interest is not the closest symmetric configuration, but rather the symmetry measure or the probability distribution of the symmetry measure values given the probability distributions of the point locations. 

Fig. 6.3. Cutoff fluorescence selection for screening. Instrumentation, labeling, and biological noise introduce spreading into a fluorescence measurement, such that the fluorescence probability distributions for wild-type and mutant cells overlap. The logarithm of single-cell fluorescence as measured by flow cytometry is generally well-approximated by a symmetrical Gaussian curve. A cutoff fluorescence value is selected for screening, with all cells above that value sorted out. The enrichment factor forthe mutants is the ratio of (dotted + striped areas)/(striped area), and the probability of retention of a given mutant clone at a single pass is the (striped + dotted area)/(all area under mutant curve).

With such a model, the ion-ion interactions are obtained by calculating the most probable distribution of ions around any central ion and then evaluating the energy of the configuration. If (r) is the spherically symmetrical potential in the solution at a distance r from a central ion i of charge z 8, then (r) will be made up of two parts ZxZlDv the coulombic field due to the central ion, and an additional part, ai(r), due to the distribution of the other ions in the solution around t. The potentials at(r) and must satisfy Poisson s equation = — 47rp/D at every point... 

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