Gaussian probability density

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)... 

In an extended version of the hopping concept, positional ( off-diagonal ) disorder in addition to energetic ( diagonal ) disorder has been introduced [54,63]. The simplest ansatz was to incorporate this by allowing the electronic overlap parameter 2ya to vary statistically. Operationally, one splits this parameter into two site contributions, each taken from a Gaussian probability density, and defines a positional disorder parameter I, in addition to the energetic disorder parameter cr. 

Let Xl9X2,...9Xr be a set of r independent stochastic variables, each having the same Gaussian probability density Px(x) with zero average and variance o2. Their sum Y has the probability density... 

Gaussian probability density distribution. Thus, when a particle passes through a turbulent eddy, we have... 

Fig. 5.8. The Gaussian probability density with mean ju. and standard deviation a.

The continuous limit of a simple random walk model leads to a stochastic dynamic equation, first discussed in physics in the context of diffusion by Paul Langevin. The random force in the Langevin equation [44], for a simple dichotomous process with memory, leads to a diffusion variable that scales in time and has a Gaussian probability density. A long-time memory in such a random force is shown to produce a non-Gaussian probability density for the system response, but one that still scales. 

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 is important to spell out the limitations on the derivation of the distribution (2.2S) of fluctuations. Consider the most general initial state, which is XPinitial state is pure if the rank of the matrix p is unity. Otherwise, it is a mixture. The transition intensity to the final state / is y = Y,ijx Pijxj where x = < T />. y = x px is then a quadratic form where the amplitudes x have a gaussian probability density... 

In the preceding equation we have assumed that the domain D is large enough to enclose all of the most important x values. We now define a normalized Gaussian probability density function in Fourier space ... 

The central limit theorem [42] shows that the normal or gaussian probability density function is a good approximation to the Poisson probability density function for large values of /u. That is,... 

The difference N in this case also has a gaussian probability density function with a population mean... 

Statistics of the Spectral Estimates, The exact results for the statistics of the AR spectral estimator are not known. For large samples of stationary processes, the spectral estimates have approximately a Gaussian probability density function and are asymptotically unbiased and consistent estimates of the power spectral density. TTie variance of the estimate is given by... 

Once the parameters of the Gaussian probability density functions for all classes are known, the density at any location can be calculated and an unknown pattern can be classified by the Bayes rule or by the maximum likelihood method. A binary classification with equal covariance matrices for both classes can be reduced in this way to a linear classifier C87, 317, 396D. 

Analytical procedures for calculating the non-Gaussian probability density function of the response are generally based on the assumption of Markov processes. Therefore, in a preceding step, the equation of motion (6) has to be transformed by methods of classical mechanics into a set of first order differential equations. From this equation a parabolic partial differential equation, the so-called Fokker-Planck-equatlon, can be derived ... 

Assuming that the velocity distribution function follows a normal distribution about its mean, the Gaussian probability density function for scalar properties is expressed as ... 

Consequently, evaluation of the Gaussian probability densities and summation (integration) over segment indices gives A 2 in the form... 

The state is described by a Gaussian probability density with zero mean and a variance, a (t), that increases with time, eventually approaching the large time asymptote. 

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