Gaussian random process

The mean values of the. (t) are zero and each is assumed to be stationary Gaussian white noise. The linearity of these equations guarantees that the random process described by the a. is also a stationary Gaussian-... 

For the usual accurate analytical method, the mean f is assumed identical with the true value, and observed errors are attributed to an indefinitely large number of small causes operating at random. The standard deviation, s, depends upon these small causes and may assume any value mean and standard deviation are wholly independent, so that an infinite number of distribution curves is conceivable. As we have seen, x-ray emission spectrography considered as a random process differs sharply from such a usual case. Under ideal conditions, the individual counts must lie upon the unique Gaussian curve for which the standard deviation is the square root of the mean. This unique Gaussian is a fluctuation curve, not an error curve in the strictest sense there is no true value of N such as that presumably corresponding to a of Section 10.1—there is only a most probable value N. 

The Shot Noise Process.—In this and the next section we shall discuss two specific random processes—the shot noise process53 and the gaussian process. These processes play a central role in many physical applications of the theory of random processes as well as being of considerable theoretical interest in themselves. 

Equation (3-325), along with the fact that Y(t) has zero mean and is gaussian, completely specifies Y(t) as a random process. Detailed expressions for the characteristic function of the finite order distributions of Y(t) can be calculated by means of Eq. (3-271). A straightforward, although somewhat tedious, calculation of the characteristic function of the finite-order distributions of the gaussian Markov process defined by Eq. (3-218) now shows that these two processes are in fact identical, thus proving our assertion. 

Radiation field, interaction with nega-ton-positon field, 642 Radtke, M. ( ., 408 Raiffa, Howard, 314 Random coding, 227 Random particle velocity, 19 Random processes, 99,102 Gaussian, 176 harmonic analysis of, 180... 

A. N. Malakhov, Cumulant Analysis of Random Non-Gaussian Processes and Its Transformations, Sovetskoe Radio, Moscow, 1978, in Russian. 

Due to its large discrete jumps, rotation around the helix axis is not a continuous Gaussian random process, so its actual mean squared angular displacement cannot be used directly in C (t). However, <<5z(/)2> for the equivalent Gaussian random process that makes the same contribution to the FPA is obtained from the relation... 

The mean velocity components are expressed as //, v, and ii>. We assume that the velocity components are stationary Gaussian random processes, so that, based on the preceding discussion, the autocovariances of u, V, and w can be written as (Papoulis, 1965, p. 397)... 

While radioactive decay is itself a random process, the Gaussian distribution function fails to account for probability relationships describing rates of radioactive decay Instead, appropriate statistical analysis of scintillation counting data relies on the use of the Poisson probability distribution function ... 

In order to treat thermal effects on small time scales, a random thermal field, Hth, is added to the effective field in the Landau-Lifshitz Gilbert equation. The thermal field is a Gaussian random process with the following statistical properties... 

To summarize, the observation of a Gaussian profile usually implies that transport is governed mathematically by the diffusion equations and mechanistically by one or more multistep random processes. Below we examine some of the random mechanisms operative in separations. 

In Eq. (22), the Langevin force F(t) may be considered as a Gaussian stationary random process of zero mean with correlation function given by Eq. (20). 

When the random process is Gaussian, we have p = 1/2. As we mentioned earlier, there are regions of size (or length /) separated by energy barriers in an amorphous solid. This size is proportional to the timescale (A) needed for a hole to penetrate a barrier of height (A) [16],... 

Figure 10. The entropy for an uncorrelated Gaussian random process generating random walk trajectories calculated using DEA is graphed versus the natural logarithm of the time. The data indicate a linear relationship between S(t) and In t as predicted by Eq. (91).

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