Gaussian random variable

If all the resonance states which fomi a microcanonical ensemble have random i, and are thus intrinsically unassignable, a situation arises which is caWtA. statistical state-specific behaviour [95]. Since the wavefunction coefficients of the i / are Gaussian random variables when projected onto (]). basis fiinctions for any zero-order representation [96], the distribution of the state-specific rate constants will be as statistical as possible. If these within the energy interval E E+ AE fomi a conthuious distribution, Levine [97] has argued that the probability of a particular k is given by the Porter-Thomas [98] distribution... 

Characterization of Chance Occurrences To deal with a broad area of statistical apphcations, it is necessary to charac terize the way in which random variables will varv by chance alone. The basic-foundation for this characteristic is laid through a density called the gaussian, or normal, distribution. 

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)... 

One consequence of Eq. (3-185) is the important result that the sum of a family of statistically independent, gaussianly distributed random variables is again gaussianly distributed. To show this, let... 

Independent gaussian random variables are by no means the only ones whose distributions are preserved under addition. Another example is independent, Poisson distributed random variables, for which... 

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—... 

We conclude this section by deriving an important property of jointly gaussian random variables namely, the fact that a necessary and sufficient condition for a group of jointly gaussian random variables 9i>- >< to be statistically independent is that E[cpjCpk] = j k. Stated in other words, linearly independent (uncorrelated),46 gaussian random variables are statistically independent. This statement is not necessarily true for non-gaussian random variables. 

The next step in the argument will be to make it plausible (a strict proof involves some limit arguments that would take us too far afield) that the (one-dimensional) distribution function of the random variable 7(f) is gaussian. To do so, we argue that the integral in Eq. (3-273) can be approximated as closely as desired by a sum of the form... 

Additive Gaussian Noise Charmed.17 An example of the use of these bounds will now be helpful. Consider a channel for which tire input is an arbitrary real number and the output is the sum of the input and an independent gaussian random variable of variance a3. Thus,... 

If U0 and U1 were the functions of a sufficient number of identically distributed random variables, then AU would be Gaussian distributed, which is a consequence of the central limit theorem. In practice, the probability distribution Pq (AU) deviates somewhat from the ideal Gaussian case, but still has a Gaussian-like shape. The integrand in (2.12), which is obtained by multiplying this probability distribution by the Boltzmann factor exp (-[3AU), is shifted to the left, as shown in Fig. 2.1. This indicates that the value of the integral in (2.12) depends on the low-energy tail of the distribution - see Fig. 2.1. 

The random displacement SqR is a Gaussian random variable with zero mean and variance... 

According to the central limit theorem, if one sums up random variables which are drawn from any (but the same for all variables) distribution (as long as this distribution has finite variance), then the sum is distributed according to a Gaussian. In this... 

Just as in everyday life, in statistics a relation is a pair-wise interaction. Suppose we have two random variables, ga and gb (e.g., one can think of an axial S = 1/2 system with gN and g ). The g-value is a random variable and a function of two other random variables g = f(ga, gb). Each random variable is distributed according to its own, say, gaussian distribution with a mean and a standard deviation, for ga, for example, (g,) and oa. The standard deviation is a measure of how much a random variable can deviate from its mean, either in a positive or negative direction. The standard deviation itself is a positive number as it is defined as the square root of the variance ol. The extent to which two random variables are related, that is, how much their individual variation is intertwined, is then expressed in their covariance Cab ... 

Figure 16.6 Calibration of the radiocarbon ages of the Cortona and Santa Croce frocks the software used[83] is OxCal v.3.10. Radiocarbon age is represented on the y axis as a random variable normally distributed experimental error of radiocarbon age is taken as the sigma of the Gaussian distribution. Calibration of the radiocarbon agegivesa distribution of probability that can no longer be described by a well defined mathematical form it is displayed in the graph as a dark area on the x axis...

Hint this is most easily done by projecting each distribution into one dimension, where it becomes a Gaussian, and using the theorem from statistics that the variance of the difference (or sum) of two random variables is the sum of the individual variances. 

Here 4 is the target state vector at time index k and Wg contains two random variables which describe the unknown process error, which is assumed to be a Gaussian random variable with expectation zero and covariance matrix Q. In addition to the target dynamic model, a measurement equation is needed to implement the Kalman filter. This measurement equation maps the state vector t. to the measurement domain. In the next section different measurement equations are considered to handle various types of association strategies. 

The vector nk describes the unknown additive measurement noise, which is assumed in accordance with Kalman filter theory to be a Gaussian random variable with zero mean and covariance matrix R. Instead of the additive noise term nj( in equation (20), the errors of the different measurement values are assumed to be statistically independent and identically Gaussian distributed, so... 

For Gaussian random variables, an extensive theory exists relating the joint, marginal, and conditional velocity PDFs (Pope 2000). For example, if the one-point joint velocity PDF is Gaussian, then it can be shown that the following properties hold ... 

The most common choice is for the components of Z to be uncorrelated standardized Gaussian random variables. For this case, ez z) = z = diag(szj,. .., szNs), i.e., the conditional joint scalar dissipation rate matrix is constant and diagonal. 

Coalescence Growth Mechanism. Following the very early step of the growth represented by Eq. (1), many nuclei exist in the growth zone. Hence Eq. (2) would be a major step for the crystal growth. Since there are many nuclei and embryos with various sizes in the zone, Uy in Eq. (2) can be assumed to be a random variable. Due to mathematical statistics, the fraction of volume approaches a Gaussian after many coalescence steps (3). A lognormal distribution function is defined by... 

Because all higher moments of fm(t) are determined by these two, when / (/) is a gaussian random variable, we do not have to specify them separately. Lax gives the general formula. 

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... 

It would be very convenient to know whether or not the linear and angular momenta can be accurately represented by stationary Gaussian random variables. If they are, then the probability of finding a molecule at time t with a velocity V, given that it was moving with a velocity V0 at the initial t = 0, is the Gaussian transition probability... 

Now, perform a cumulant expansion of the average involved in the ACF (158). Then, since the random variable is Gaussian, the expansion of Eq. (158) limits us to second order, so that the following relation holds ... 

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