Gaussian random function

Although Gaussian random functions are commonly encountered, their representation in terms of functional integrals is often unnecessary the averages (6.27) are usually sufficient for the development. A familiar example occurs in the description of Brownian motion by means of the Langevin equation - ... 

Note that (10.6) is the same kind of functional integral that is introduced in (6.34) in an entirely different context. It is also shown to represent the variance of a Gaussian random function as encountered in the random forces in Brownian motion. 

The Maxwell-Boltzmann velocity distribution function resembles the Gaussian distribution function because molecular and atomic velocities are randomly distributed about their mean. For a hypothetical particle constrained to move on the A -axis, or for the A -component of velocities of a real collection of particles moving freely in 3-space, the peak in the velocity distribution is at the mean, Vj. = 0. This leads to an apparent contradiction. As we know from the kinetic theor y of gases, at T > 0 all molecules are in motion. How can all particles be moving when the most probable velocity is = 0 ... 

This result checks with our earlier calculation of the moments of the gaussian distribution, Eq. (3-66). The characteristic function of a gaussian random variable having an arbitrary mean and variance can be calculated either directly or else by means of the method outlined in the next paragraph. 

Equation (3-88) enables us to calculate the characteristic function of the unnormalized random variable from a knowledge of the characteristic function of . For example, the characteristic function of a gaussian random variable having arbitrary mean and variance can be written down immediately by combining Eqs. (3-83) and (3-88)... 

The central limit theorem thus states the remarkable fact that the distribution function of the normalized sum of identically distributed, statistically independent random variables approaches the gaussian distribution function as the number of summands approaches infinity—... 

For most systems in thermal equilibrium, it is sufficient to regard fB as random forces which follows a Gaussian distribution function with mean value = 0 and standard deviation = 2kBT 8(i — j) 8(t — t ) [44],... 

A classical description of such a structure is of no real use. That is, if we attempt to describe the structure using the same tools we would use to describe a box or a sphere we miss the nature of this object. Since the structure is composed of a series of random steps we expect the features of the structure to be described by statistics and to follow random statistics. For example, the distribution of the end-to-end distance, R, follows a Gaussian distribution function if counted over a number of time intervals or over a number of different structures in space,... 

Here, 7 is the friction coefficient and Si is a Gaussian random force uncorrelated in time satisfying the fluctuation dissipation theorem, (Si(0)S (t)) = 2mrykBT6(t) [21], where 6(t) is the Dirac delta function. The random force is thought to stem from fast and uncorrelated collisions of the particle with solvent atoms. The above equation of motion, often used to describe the dynamics of particles immersed in a solvent, can be solved numerically in small time steps, a procedure called Brownian dynamics [22], Each Brownian dynamics step consists of a deterministic part depending on the force derived from the potential energy and a random displacement SqR caused by the integrated effect of the random force... 

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 ... 

Thus the Aq are independent Gaussian random variables with zero mean. Accordingly u(, t) has become a random field, i.e., a random function of the four variables, t rather than of t alone. One is interested in its stochastic properties, for instance, the two-point correlation function... 

Random broadening effects can dominate the lineshape and lead to a lineshape that is almost Gaussian. This is often the case in MRS.32 The Gaussian lineshape is also a convenient approximation to the lineshape for some solids.10 The normalized Gaussian lineshape function in the frequency domain is given by... 

Figure 1. Optimal control between Gaussian random vectors in a 64 x 64 random matrix system by the Zhu-Botina-Rabitz scheme with T — 20 and a = 1. a) The optimal field after 100 iterations b) its power spectrum (c) the optimal evolution of the squared overlap with the target (([)(r) (py) as well as its magnified values near the target time in the inset (d) the convergence behavior of the overlap Jq (solid curve) and the functional J (dashed curve) versus the number of iteration steps.

The majority of statistical tests, and those most widely employed in analytical science, assume that observed data follow a normal distribution. The normal, sometimes referred to as Gaussian, distribution function is the most important distribution for continuous data because of its wide range of practical application. Most measurements of physical characteristics, with their associated random errors and natural variations, can be approximated by the normal distribution. The well known shape of this function is illustrated in Figure 1. As shown, it is referred to as the normal probability curve. The mathematical model describing the normal distribution function with a single measured variable, x, is given by Equation (1). 

Equations (7.60)-(7.63) describe general properties of many-variable Gaussian distributions. For a Gaussian random process the set zy corresponds to a sample zy,Z, from this process. This observation can be used to convert Eq. (7.63) to a general identity for a Gaussian stochastic process z(t) and a general function of... 

The identities (7.63) and (7.64) are very useful because exponential functions of random variables of the forms that appear on the left sides of these identities are frequently encountered in practical applications. For example, we have seen (cf. Eq. (1.5)) that the average (e ), regarded as a function of a, is a generating function for the moments of the random variable z (see also Section 7.5.4 for a physical example). In this respect it is useful to consider extensions of (7.63) and (7.64) to non-Gaussian random variables and stochastic processes. Indeed, the identity (compare Problem 7.8)... 

Using this result in Eq. (8.14) we find that the correlation function of the Gaussian random force R has the form... 

The above equation is the characteristic function of a one-dimensional Gaussian random variable having mean... 

In this work we assume that the measurement errors are Gaussian random variables, with known (modeled) means and covariances, and that the measurement errors are additive and independent of the unknowns. With these hypotheses, the likelihood function can be expressed as [28, 29] ... 

Recall that the likelihood function for a model with Gaussian random errors (the observed data) can be written as... 

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