Uncorrelated random variables

Uhlenbeck G. E., 170,176 Umezawa, ffc, 698 Uncorrelated random variables, 146 Uniform distribution, 109 Unit cell... 

According to the definition of the standard error, if is the standard error of h, it ought to have such a value that a new average h would have a 68.3 percent chance of falling between h — and h + cr. To obtain the standard error of n, consider Eq. 2.71 as a special case of Eq. 2.36a. The quantity is a linear function of the uncorrelated random variables n, n2,...,n, each with standard deviation o-. Therefore... 

If tii,n2..riff are mutually uncorrelated random variables with a common variance a, ... 

This is equivalent to the statement that the sum over a large number of subsystems gives the average value, something that would be expected in case the central limit theorem (CLT) is applicable. This generally is the case if quantities like M for the sub-blocks are independent and uncorrelated random variables. 

Consider a set of states labeled by the index a whose physical meaning will be elucidated shortly. Each of these states is characterized by a weight Wa and a position variable Rq. Given Rq, the variables R are an infinite set of uncorrelated random variables distributed according to... 

According to the Karhunen-Loeve (K-L) theorem, a stochastic process on a bounded interval can be represented as an infinite linear combination of orthogonal functions, the coefficients of which constitute uncorrelated random variables. The basis functions in K-L expansions are obtained by eigendecomposition of the autocovariance function of the stochastic process and are shown to be its most optimal series representation. The deterministic basis functions, which are orthonormal, are the eigenfunctions of the autocovariance function and their magnitudes are the eigenvalues. The Karhunen-Loeve expansion converges in the mean-square sense for any distribution of the stochastic process (Papoulis and Pillai 2002). A K-L representation of a zero-mean stochastic process f(t, 6) can be represented in the form... 

Here, the parameter t indicates time and 6 represents the random dimension. Note that the autocorrelation function Rff(t, s) need not be stationary. In Eq. 11,, (0) is a vector consisting of uncorrelated random variables with zero mean and unit variance. Since, (0 are eigenfunctions, it is obvious that... 

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 proof that the variance of the sum of two terms is equal to the sum of the variances of the individual terms is a standard derivation in Statistics, but since most chemists are not familiar with it we present it in the Appendix. Having proven that theorem, and noting that AEs and AEr are independent random variables, they are uncorrelated and we can apply that theorem to show that the variance of AT is ... 

Equation 41-A3 can be checked by expanding the last term, collecting terms and verifying that all the terms of equation 41-A2 are regenerated. The third term in equation 41-A3 is a quantity called the covariance between A and B. The covariance is a quantity related to the correlation coefficient. Since the differences from the mean are randomly positive and negative, the product of the two differences from their respective means is also randomly positive and negative, and tend to cancel when summed. Therefore, for independent random variables the covariance is zero, since the correlation coefficient is zero for uncorrelated variables. In fact, the mathematical definition of uncorrelated is that this sum-of-cross-products term is zero. Therefore, since A and B are random, uncorrelated variables ... 

Thirdly, when we separated equation 43-51 into two terms, we only worked with the first term. The second term, which we presented in equation 43-52B, was neglected. Is it possible that the nonlinear effects observed for equation 43-52A will also operate on equation 43-52B The answer is yes, it will, but... And the but..is this AEs is a random variable, just as AEr is. Furthermore, it is uncorrelated with AEx. Therefore, in order to evaluate the integral representing the variation of both AEs and AEr, it would be necessary to perform a double integration over both variables. Now, for each value of AEs, the nonlinearity caused by the presence of AEr in the denominator would apply. However, AEs is symmetrically distributed around zero, therefore for every positive value of AEs there is an equal but negative value that is subject to exactly the same nonlinear effect. The net result is that these pairs always form equal and opposite contributions to the integral, which therefore cancel, leaving no effect due to AEs. 

It has to be noted that the measurement values for range and velocity are not uncorrelated according to the LFMCW measurement described in section 8. As a consequence, the observed measurement errors ft, can also be considered as correlated random variables for a single sensor s data. For 24GHz pulse radar networks, developed also for automotive applications, a similar idea has been described by a range-to-track association scheme [12], because no velocity measurements are provided in such a radar network. 

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. 

In the theory of probability the term correlation is normally applied to two random variables, in which case correlation means that the average of the product of two random variables X and Y is the product of their averages, i.e., X-Y)=(,XXY). Two independent random variables are necessarily uncorrelated. The reverse is usually not true. However, when the term correlation applies to events rather than to random variables, it becomes equivalent to dependence between the events. 

The two random variables x andy are called independent or uncorrelated if... 

Equations 2.81 and 2.83 are the answers to questions 1 and 2 stated previously. They indicate, first, that the average of the function is calculated using the average values of the random variables and, second, that its standard error is given by Eq. 2.83. Equation 2.83 looks complicated, but fortunately, in most practical cases, the random variables are uncorrelated—i.e., p, = 0, and Eq. 2.83 reduces to... 

The conceptual idea of geostatistics is that spatial variation of any variable Z can be expressed as the sum of three major components (Equation 15.1) (i) a structural component, having a constant mean or trend that is spatially dependent, (ii) a random, but spatially correlated component, and (iii) spatially uncorrelated random noise or residual term (Webster and Oliver, 2001) ... 

Furthermore, simulation shows that the random variables Sy icok) and Sy kf ojk ) are un-correlated in the same range of frequencies for the Chi-square distribution, for k k and k, k 6 K.. According to Yaglom [277], uncorrelated Chi-square random variables are independent. Use K to denote the, frequency index set that contains the frequency indices for these approximations to be accurate. Given the observed data V, the spectral set can be computed by using Equation (3.25) ... 

If the matrix PH P is diagonal, say D, the Gaussian random variables y and y2 are uncorrelated and, hence, statistically independent. Then, it is an easy task to draw the PDF contours in the y —9 — y2 coordinate system. In order to obtain a solution for the matrix P to fulfill this goal, consider the eigenvalue problem of the covariance matrix ... 

In general when p = 0 the two random variables are said to be uncorrelated. Note that independent random variables are always uncorrelated (you should be able to simply show this). 

In the alternative, a narrowband representation could be used.) Since and are linear transformations of the Gaussian random variable n(f), they are also Gaussian random variables [7.66] furthermore it can be shown that for Tlarge, all u s and a s are uncorrelated and independent of one another [7.67]. Since the mean of n(r) is taken to be zero, we find... 

X ] = I). To this aim, the basic random variables, X, can be transformed into a set of uncorrelated standard variates U through the linear transformation ... 

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