Correlated/uncorrelated random variables

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 ... 

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. 

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 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) ... 

When samples are not discriminated, the problem of the interpretation of evidence becomes primordial and much more difficult. There is some need to know the risk or the probability that two items come from different sources, but are not discriminated. This is expressed by the discrimination power of the attribute analyzed that defines the effectiveness of that attribute to differentiate samples in forensic science. Extensive databases are used to compute the probability that two randomly selected samples will be discriminated when the attribute is analyzed or measured. This probability is, in effect, the discrimination power that has been calculated for uncorrelated, correlated, and continuous variables such as blood grouping systems, drugs, glass, paints, etc. 

There are two common assumptions made about the individual specific effect, the random effects assumption and the fixed effects assumption. The random effects assumption (made in a random effects model) is that the individual specific effects are uncorrelated with the independent variables. The fixed effect assumption is that the individual specific effect is correlated with the independent variables. If... 

The generating random process we used is based on a rather subtle mathematical technique that we cannot describe here. Basically, we start from a symmetric, positive definite, correlation matrix A from which we deduce an accessory matrix B using the Cholesky method. The required vector U whose the components are the correlated velocity fluctuations is then equal to the matrix B multiplied by a vector whose components are uncorrelated, centered, normal variables of variances unity. The procedure first designed for an lD formulation has been extended to 2D-problems. Mean turbulence inhomogeneities can be accounted for in the process. Details can be found in Desjonqu res, 1987, Berlemont, 1987, Gouesbet et al, 1987, Berlemont et al, 1987, Desjonqu res et al, 1987. 

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