Probability density function Gaussian

Figure 4 Sample spatial restraint m Modeller. A restraint on a given C -C , distance, d, is expressed as a conditional probability density function that depends on two other equivalent distances (d = 17.0 and d" = 23.5) p(dld, d"). The restraint (continuous line) is obtained by least-squares fitting a sum of two Gaussian functions to the histogram, which in turn is derived from many triple alignments of protein structures. In practice, more complicated restraints are used that depend on additional information such as similarity between the proteins, solvent accessibility, and distance from a gap m the alignment.

We conclude this section with examples of some particularly important probability density functions that will be used in later applications. In each of these examples, the reader should verify that the function px is a probability density function by showing that it is non-negative and has unit area. All of the integrals and sums involved are elementary except perhaps in the case of the gaussian distribution, for which the reader is referred to Cramer.7... 

The thermally produced noise voltage X(t) appearing across the terminals of a hot resistor is often modeled by assuming that the probability density function for X(t) is gaussian,... 

The last example brings out very clearly that knowledge of only the mean and variance of a distribution is often not sufficient to tell us much about the shape of the probability density function. In order to partially alleviate this difficulty, one sometimes tries to specify additional parameters or attributes of the distribution. One of the most important of these is the notion of the modality of the distribution, which is defined to be the number of distinct maxima of the probability density function. The usefulness of this concept is brought out by the observation that a unimodal distribution (such as the gaussian) will tend to have its area concentrated about the location of the maximum, thus guaranteeing that the mean and variance will be fairly reasdnable measures of the center and spread of the distribution. Conversely, if it is known that a distribution is multimodal (has more than one... 

It can be shown that the right-hand side of Eq. (3-208) is the -dimensional characteristic function of a -dimensional distribution function, and that the -dimensional distribution function of afn, , s n approaches this distribution function. Under suitable additional hypothesis, it can also be shown that the joint probability density function of s , , sjn approaches the joint probability density function whose characteristic function is given by the right-hand side of Eq. (3-208). To preserve the analogy with the one-dimensional case, this distribution (density) function is called the -dimensional, zero mean gaussian distribution (density) function. The explicit form of this density function can be obtained by taking the i-dimensional Fourier transform of e HsA, with the result.45... 

Fig. 4.4. Probability density function of the force. The mean is — 1.1 and the standard deviation is 13.2. A fit with a Gaussian distribution with identical mean and variance is shown...

Gao, F. (1991). Mapping closure and non-Gaussianity of the scalar probability density function in isotropic turbulence. Physics of Fluids A Fluid Dynamics 3, 2438-2444. 

A normal (gaussian) probability density function in one centered and standardized variable X reads... 

The Gaussian concept can be extended beyond that already developed in Section IV. The general Gaussian probability density function for the position of a fluid particle released from a source located at (x y, z ) at time t can be expressed as (Lamb, 1980)... 

Fig. 33. Probability density function for Gaussian bivariate distribution...

A very important probability distribution is the normal or Gaussian distribution (after the German mathematician, Karl Friedrich Gauss, 1777-1855). The normal distribution has the same value for the mean, median and mode. The equation describing this distribution (the probability density function)... 

The normal, or Gaussian, distribution occupies a central place in statistics and measurement. Its familiar bell-shaped curve (the probability density function or pdf, figure 2.1) allows one to calculate the probability of finding a result in a particular range. The x-axis is the value of the variable under consideration, and the y-axis is the value of the pdf. 

One powerful technique is Maximum Likelihood Estimation (MLE) which requires the derivation of the Joint Conditional Probability Density Function (PDF) of the output sequence [ ], conditional on the model parameters. The input e n to the system shown in figure 4.25 is assumed to be a white Gaussian noise (WGN) process with zero mean and a variance of 02. The probability density of the noise input is ... 

A more fruitful solution to the closure problem is provided by the use of probability density functions for the fluctuating components. Various shapes (spiked, square wave, gaussian distributions) have been successfully tried (3). 

The expression of the probability density function, P r), in logarithmic form corresponding to non-Gaussian statistics furnished by Kuhn and Griin in 1942 for a chain with n links of length / is given by... 

The energetics of such atomic motion can be investigated. If the probability density function is a Gaussian function, the potential energy in which the atom vibrates will be isotropic and harmonic and will have a normal Boltzmann distribution over energy levels. This potential energy will have the form ... 

The Gaussian function is frequently used as the weight function because it is well behaved, easily calculated, and satisfies the conditions required by Parzen s estimator. Thus the probability density function for the multivariate... 

This review has mainly focused on models for mean concentrations. However, fluctuations need to be estimated in order to assess all the risks associated with accidental releases. There is some evidence from the experiments, Davidson et al., 1995 [143], that the intensity of fluctuations is lower in clouds/plumes released among buildings, and are also qualitatively different. There is much less chance of a large scale of wind gust reducing the concentration to zero, so that the probability density function is closer to a log-normal distribution than to a cut-off Gaussian (Mylne, 1992 [440]). 

The tails prevent [19] convergence to the Gaussian distribution for N -= oo, but not the existence of a limiting distribution. These distributions as we have seen are called stable distributions. If the concept of a Levy distribution is applied to an assembly of temporal random variables such as the x, of the present chapter, then w(x) is a long-tailed probability density function with long-time asymptotic behavior [7,37],... 

ADAPT-LODI, developed at Lawrence Livermore National Laboratory. The ADAPT model assimilates meteorological data provided by observations and models (in particular, by Coupled Ocean/Atmosphere Mesoscale Prediction System [COAMPS ]) to construct the wind and turbulence fields. Particle positions are updated using a Lagrangian particle approach that uses a skewed (non-Gaussian) probability density function (Nasstrom et al. 1999 Ermak and Nasstrom 2000). 

It was further noticed that the value of the coefficient was sensitive to the density ratio between the continuous and dispersed phases, [p /p ). Moreover, in a recent study Brenn et al [9] investigated unsteady bubbly flow with very low void fractions and concluded that the velocity probability density functions of bubbles in liquid are better described using two superimposed Gaussian functions. 

Given the dependency of the wavelet coefficients, one still has to find the appropriate framework for modeling their probability density functions. A Gaussian model is not appropriate since the wavelet decomposition tends to produce a large number of small coefficients and a small number of... 

From a practical point of view the classical method using directly the probability density function is not convenient, and it is computationally preferable to use an approach that involves trajectory calculations. A derivation of such formulation can be made by starting from the quantum-mechanical TDSCF, and using semiclassical (gaussian) wavepackets. Here we merely quote the final result. In analogy to (62), the single-mode classical SCF potentials are given by... 

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