Linear combinations of random variables

The expected value can now be found through a linear combination of random variables... 

In the collapse phase the monomer density p = N/R is constant (for large N). Thus, the only confonnation dependent tenn in (C2.5.A1) comes from the random two-body tenn. Because this tenn is a linear combination of Gaussian variables we expect that its distribution is also Gaussian and, hence, can be specified by the two moments. Let us calculate the correlation i,) / between the energies and E2 of two confonnations rj ]and ry jof the chain in the collapsed state. The mean square of E is... 

Principal Components (i. e, PCs) are linear combinations of random or statistical variables, which have special properties in terms of variances. 

This relation is legitimate, since both the observed responses and the predicted values are random variables. This R value, which is called the multiple correlation coefficient, is never negative. It is the largest correlation that a linear combination of independent variables, in the form specified by the model, could have with the observed y values. 

Principal components are linear combinations of random or statistical variables, which have special properties in terms of variances. The central idea of PCA is to reduce the dimensionality of a data set that may consist of a large number of interrelated variables while retaining as much as possible of the variation present in the data set. This is achieved by transforming the PCs which are uncorrelated into a new set of variables which are ordered so that the first few retain most of the variation present in all of the original variables [292-295]. 

For independent random variables, the covariances are zero, thus the following formula holds for the variance of the linear combination of several variables ... 

Consider next the problem of estimating the error in a variable that cannot be measured directly but must be calculated based on results of other measurements. Suppose the computed value Y is a linear combination of the measured variables yt], Y = oqt/1 + a2y2 + . Let the random variables y2, have means E(yi), E(y2),. . . and variances G2(yi), G2(t/2),... The variable Y has mean... 

Following the random-function model (1), consider the prediction of T(x ) by f (.v) = a x)y, that is, a linear combination of the n values of the output variable observed in the experiment. The best linear unbiased predictor is obtained by minimizing the mean squared error of the linear predictor or approximator, Y(x). The mean squared error, MSIi K(x), is... 

Thus the random variables Z that compose Z are independent and, being linear combinations of Xjt they are of normal distribution. The symmetry measure, as defined in Section III, is equivalent, in the current notations, to S = Y Y. Having S orthonormal we have ... 

In this notation, g(9, zj) is used to represent a function (g), perhaps a linear combination of CO variates, that describes the expectation of the /th subjects parameter vector 9i conditional on their demographic characteristics (z,) and population parameter values (0). The variance-covariance matrix (Q) therefore describes the random variability between subjects that is not able to be explained by covariates. 

If a random experimental error arises as the sum of many contributions, the central limit theorem of statistics gives some justification for assuming that our experimental error will be governed by the Gaussian distribution. This theorem states that if a number of random variables (independent variables) x, X2,..., x are governed by some probability distributions with finite means and finite standard deviations, then a linear combination (weighted sum) of them... 

The estimators of the coefficients (bo, b, and bf) are random variables obtained by linear combinations of experimental values, themselves random variables. There is uncertainty in the calculated response y point A in the domain, defined by its variance, var(y ). This in turn depends on the experimental variance o, which is assumed constant, and the variance function d at that point ... 

It is worth noting at this point that expressions (2.46) and (2.47), taken together, inform us that the paths x are linear combinations of Gaussian-distributed random variables a, as pointed out by Coalson et al. [39]. 

The expected values over functions and linear combinations of the random variable are... 

Equation (3.31) indicates that x t)—x is a linear combination of Gaussian random variables f t). Therefore, according to the theorem in Appendix 2.1, the distribution of x(t) must be Gaussian. Hence the probability distribution of x(t) is written as... 

First we consider the Rouse model, for which g(k, t) can be calculated rigorously. According to the theorem in Appendix 2.1, a linear combination of the Gaussian random variables obeys the distribution. Since R (t) — R (0) is a linear function of which is Gaussian, the distribution of l (t) —1 (0) is also Gaussian. Hence, eqn... 

---