Marginal density function

An important concept is the marginal density function which will be better explained with the joint bivariate distribution of the two random variables X and Y and its density fXY(x, y). The marginal density function fxM(x) is the density function for X calculated upon integration of Y over its whole range of variation. If X and Y are defined over SR2, we get... 

It is readily seen that the transform of the marginal density function is given by ... 

Mandelstam, S., 356,371,377,381,664 Mandelstam s postulate, 376 Many particle state, 540 Margenau distribution, 49 Margenau, Henry, 49,391 Marginal densities of probability density functions, 138... 

Figure 4.8 A bivariate probability density function. The slice parallel to the y axis represents the marginal density fxM(x).

Let us now turn to the density functional methods. All of them correctly predict the para-ortho ordering, but considering that this is the case even for a Hartree-Fock treatment this is a somewhat hollow victory. Without exception, all functionals wrongly predict p-protonation. Arguably, this small energy difference falls within the error margin of any type of calibration for (semi-) empirical DFT functionals. 

The electron and momentum densities are just marginal probability functions of the density matrix in the Wigner representation even though the latter, by the Heisenberg uncertainty principle, cannot be and is not a true joint position-momentum probability density. However, it is possible to project the Wigner density matrix onto a set of physically realizable states that optimally fulfill the uncertainty condition. One such representation is the Husimi function [122,133-135]. This seductive line of thought takes us too far away from the focus of this... 

Under the assumption that the random variables are independent, this probability is given by the product of the marginal probability density functions integrated over the region a>s, i.e.,... 

A random variable with a value denoted by is said to be statistically independent from a random variable with a value denoted by V2 if P(vi v2) is independent of for all 1)2—that is, if the conditioning does not affect the probability-density function. In this case P(v v2) — P(rJ—that is, the conditioned probability-density function is the same as the marginal probability-density function. From equation (12) it then follows that... 

It would be possible to estimate 6 and cr jointly via Eq. (6.6-1) however, this is not the usual procedure. The standard procedure is to integrate Eq. (6.1-13) over the permitted range of cr, thus obtaining the marginal posterior density function... 

Quite a different approach was undertaken by Ahlrichs and Elliott88 who studied AlN clusters using an accurate density-functional approach. Due to the computational demands of the electronic-structure approach, only the smallest clusters could be studied and for those it was not possible to perform a complete structure optimization. In addition, the authors studied also selected, high-symmetric clusters for larger N. Among others, they compared their results with the predictions of the jellium model and found only a very marginal... 

Table 13. Experimental values for the molar susceptibility in binary B32-type compounds in units of 10 cm /mol. The first two columns are the values of Klemm and Fricke . These are average values of 48 to 52 at. pet, of Li, The third column contains the values of Yao These are interpolated values for the composition of 50 at, pet. Li. The 4th column contains the contribution of the core electrons gained for density function calculations for free atoms. In the 5th and 6th columns the contributions of the valence electrons are listed Xi(val) = Xeip.i - Xmre and Xsfval) = Xexp,j Xcote- (Numbers in parentheses indicate error margins.)...

Two techniques for dealing with these challenges, effective core potentials (or pseudopotentials) and density functional theory, have quickly transformed themselves from marginal techniques, once primarily the domain of solid-state chemists and physicists, to almost de rigueur standards for the computational... 

Now, if we take (10.4) multiply it by the marginal probability density function of the mean in the placebo group and integrate we shall obtain the probability that all n contrasts will be significant. This is given by... 

Finally, we stress that our study does not answer all questions. Thus, the fact that we use an approximate density functional in our parameter-free calculations may be one source of errors in the calculated quantities, although this problem is only marginally related to that of a proper treatment of the external field in an electronic-structure method. Second, our method is still in its infancy and many tests are required before it can be established whether it is a useful approach. Third, we have presented a method for directly including a DC field in the calculations, whereas other approaches, based on perturbation theory, also allow for the treatment of AC fields. Our approach allows for an alternative control of the results of the latter in the limit of vanishing frequencies, but it still is an open question how the results can be used in improving the perturbation-theoretic approaches. 

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