Gaussian joint probability

It can be shown that the right-hand side of Eq. (3-208) is the -dimensional characteristic function of a -dimensional distribution function, and that the -dimensional distribution function of afn, , s n approaches this distribution function. Under suitable additional hypothesis, it can also be shown that the joint probability density function of s , , sjn approaches the joint probability density function whose characteristic function is given by the right-hand side of Eq. (3-208). To preserve the analogy with the one-dimensional case, this distribution (density) function is called the -dimensional, zero mean gaussian distribution (density) function. The explicit form of this density function can be obtained by taking the i-dimensional Fourier transform of e HsA, with the result.45... 

Since the joint probability density of the complete set of positions and momenta is Gaussian, the distribution of any subset must also be Gaussian. But the characteristic function corresponding to a Gaussian density takes a simple form, and in the present case is... 

The information theory approach to calculating approximate probabilities is quite general and, as we have just shown, is quite straight forward to use. One might then ask why we did not use this approach in the previous section to predict e2(0> e4(7),. .., e2J(t), and e4J(f) from and Aj(t) The answer to this question is that we did. That is, information theory predicts Gaussian transition probabilities for V and J and these were the transition probabilities that we assumed. We shall now elaborate on this remark. Let P(V, t V0,0) be the joint probability that a molecule has a velocity V at time t and a velocity V0 at t = 0. then P is related to the transition probability Pv by... 

Consider a Gaussian random vector 0 with mean 0 and covariance matrix Z so its joint probability density function (PDF) is given by ... 

Thus the prior distribution of f is iV(0, Q) = Ai(0, fflRR ). However, each measurement contains noise, which we assume to be Gaussian with zero mean and variance cr. The vector of data points also has Gaussian distribution E(t) A (0, Q + ffv). We denote the covariance matrix of t by C =Q + ffyl-The distribution of the joint probability of observing tjv+i having previously observed t can be written as... 

The function 0(Vj) in eq. (16) represents a N(0,1) standard normal PDF, 0(V, V2,V3 R ) a 3-d1mens1onal Gaussian joint distribution with zero mean and unit standard deviations and 4>(.) the standard cumulative normal probability. All components of the correlation matrix [pjj]... 

First, consider the generic performance function G(X), and let fx(x) denote the joint probability density function of X. Recall X = X / = 1 to n], and let fix. and crx, denote the mean and standard deviation of respectively. Further, the covariance of X, and Xj is denoted by Cov(X Xj). The first-order second-moment (shortly, referred to as FOSM) method approximates G to be a Gaussian distribution, using only the mean and covariance of X. 

The likelihood function is derived from the probability distribution of the measurement errors relating to the diagnostic functionals. An appropriate distribution is a multivariate Gaussian with independent errors. Thus the joint probability density of the diagnostics and the input is... 

The meaning of the vector R,- becomes clear from Figure 12. In a macroscopically large sample, the number of polymers is sufficiently large, to describe the polymer conformation with the probability distribution of eqn [18]. The average value of the phase factors between the positions i and j within the chain follows from the approximate Gaussian distance probability eqn [18] of any seaion of the freely jointed chain ... 

Derivation of the Gaussian Distribution for a Random Chain in One Dimension.—We derive here the probability that the vector connecting the ends of a chain comprising n freely jointed bonds has a component x along an arbitrary direction chosen as the x-axis. As has been pointed out in the text of this chapter, the problem can be reduced to the calculation of the probability of a displacement of x in a random walk of n steps in one dimension, each step consisting of a displacement equal in magnitude to the root-mean-square projection l/y/Z of a bond on the a -axis. Then... 

The search for the form of W of vulcanized rubbers was initiated by polymer physicists. In 1934, Guth and Mark2 and Kuhn3) considered an idealized single chain which consists of a number of links jointed linearly and freely, and derived the probability P that the end-to-end distance of the chain assumes a given value. The resulting probability function of Gaussian type was then substituted into the Boltzmann equation for entropy s, which reads,... 

This assumption is equivalent to considering the polymer molecule as a Gaussian chain. For a Gaussian chain the probability of the two ends colliding in three-dimensional space is proportional to its length to the power -3/2. For the Kuhn (or freely-jointed chain) model the same assumption maybe taken for sufficiently long chains [60]. For linear polymers in good solvents, no similar simple assumption can be adopted. To study cyclization one has to resort to more sophisticated mathematical treatments (see, e.g. [61]). 

One powerful technique is Maximum Likelihood Estimation (MLE) which requires the derivation of the Joint Conditional Probability Density Function (PDF) of the output sequence [ ], conditional on the model parameters. The input e n to the system shown in figure 4.25 is assumed to be a white Gaussian noise (WGN) process with zero mean and a variance of 02. The probability density of the noise input is ... 

The problem considered here is the estimation of the state vector X (which contains the unknown parameters) from the observations of the vectors = [yo> yi.yk ] Because the collection of variables Y = (yoYi - -yk) jointly gaussian, we can estimate X by maximizing the likelihood of conditional probability distributions p(Xk/Yk), which are given by the values of conditional variables. Moreover, we can also search the estimate X, which minimizes the mean square error k = Xk — Xk. In both cases (maximum likelihood or least squares), the optimal estimate for the jointly gaussian variables is the conditional mean and the error in the estimate is the conventional covariance. 

At equilibrium, the distribution of conformations in a solvent at the theta temperature (see Section 2.3.1.2), or in a concentrated solution, is given by a set of random walks or, equivalently, by the conformations of a. freely jointed chain (see Section 2.2.3.2). If one end of the freely jointed chain with links, each of length bjc, lies at the origin, then-the probability, jrodR, that the other end lies at a position between R and R + dR is approximately a Gaussian function (Flory 1969 Larson 1988) ... 

Thus, as given by Eq. (1.42), the probability distribution function for the end-to-end vector R is Gaussian. The distribution has the unrealistic feature that R can be greater than the maximum extended length Nb of the chain. Although Eq. (1.42) is derived on the freely jointed chain model, it is actually valid for a long chain, where the central limit theorem is applicable, except for the highly extended states. 

Consider a vector of two Gaussian random variables 0 = [6, 62V with mean 0 = [0J, l and covariance matrix E. The goal here is to obtain the parametric form of the joint PDF contour that covers an area with a prescribed probability. First, define a vector of two new random variables y = [yi, y2] by the following transformation ... 

---