Joint density function

The joint density function for each voxel can be reconstructed by taking inverse Fourier transforms with respect to each of the wave vectors ... 

Two random variables X and Y are independent if their joint density function fXY can be factored as a product of two density functions, each involving one variable, e.g.,... 

Fig. 3 Calculated energies of solvation v. experimental values for (from left to right) water, ethanol, methanol, and methane in water. The diagonal line gives perfect agreement, the open circles results with the joint density-functional theory for electronic structure of solvated systems, and the other symbols results from different continuum approaches. Reproduced with permission from ref. 54. 

In order to obtain spatial resolution of the molecular translations within each voxel, it is necessary to combine the velocity-encoding gradient sequence shown in Section 3.1.1 with a standard imaging sequence such as frequency or phase encoding. The displacement distribution function, P A (R), must now be generalized to a spatial-displacement joint density function Aso that it describes the total number of spins located at r with displacement R during time A 28... 

An example of a joint density function gA (r, R) is shown in Fig. 12. The sample is a Bentheimer sandstone in a rectangular parallelepiped shape 50 mm long extending in the -direction, 25 mm wide along the x-direction, and 5 mm thick in the y-direction. The average volumetric flow rate of the water was 1.5 ml min-1 along the -direction. The sample is located between the two spikes resulting from free water present in the end caps of the core holder. 

Main objective in this paper is the revision and application of parametric methods in imcertainty propagation. However, before applying these methods we must ensure that output vector really follows a normal multivariate distribution. If no information is available about the joint density function in the output, the widespread procedure to ensure this assumption is by means of a multinormal contrast of goodness of fit. Fortunately, following theorem ensures the as3nnptotic j oint normal distribution o f the output vector, when this normality is fulfilled in the input vector, under some weakly conditions about the differentiability of functions in the simulator, providing also the mean vector and covariance matrix of the output vector as functions of their equivalents parameters in the input and the partial derivates of functions in the simulator. 

The structural reliability problem seeks the estimation of the probability that a structure exceeds a critical state defined by a state function indexed by a vector of so-called basic variables X, which obeys a joint density function fx X). Hence, the problem is written as follows ... 

For a single random variable the maximum entropy distribution is obtained by considering only the moment constraints. For the multivariate distribution the correlations between each pair of random variables has to be taken into account as well. This would lead to an optimization problem with 4 optimization parameters with 4 constraints from the marginal moment conditions and ( 4-1)/2 constraints from the correlation conditions, where n is number of random variables. This concept was recently apphed in Soize (2008) to determine the joint density function. For a... 

In the case of continuous distributions, independence also means that the joint density function is equal to the product of the individual density functions ... 

Figure 33.10 plots the pairs of scores of every two eigenfunctions for the first three ones. The variables Zi and Z2 are strongly correlated and therefore, are not independent. It is possible, however, to infer their joint density functions from the samples of values of the scores as shown in Fig. 33.11. 

This is analogous to Equation 5.11. When the continuous state space has dimension p, the integrals are multiple integrals over p dimensions, and the joint density function is found by taking p partial derivatives. 

However, in a general structural reliability analysis problem, the input to G is a vector, i.e., X = [Xi, X2,. . -Xj,. . Jf , and each X, has its own mean px and standard deviation maximum likelihood, which can be calculated by maximizing the joint probability density function of all the input random variables. If the variables are independent, then the joint density function of X is expressed as (when the variables are independent)... 

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