Marginal probability density

Then the two sets are called statistically independent of each other. The factor Ps is then also the marginal probability density of the variables Xl9 Xl9..., Xs. At the same time it is the conditional probability density... 

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... 

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... 

Assuming statistical independence of marginal probability density functions f t) and/( ), the equation of maintenance task completion probability is... 

The marginal probability density function represents the probability for a subset of random variables in the original joint probability density function. The subset considered is used as the subscript for the function, for example,/x(x), would be the marginal probability density function for X. The marginal probability density function is obtained by integrating out all the remaining variables, that is. 

When dealing with a multivariate distribution, the computation of the marginal and conditional probabilities is more complex. Let D = (Z>i,Z)2, , Om) be an OT-dimensional subset of the n-dimensional vector X, and let d be defined similarly. Let Xy be defined such that it contains all the variables in X that are not in D and let Xy be defined similarly. Let D be the subset of for the D vector. The marginal probability density function can then be written as... 

The marginal probability density Eii(ti)dti is the fraction of the outflow that has spent time t to t + dt in the i-th environment irrespective of the times spent in other environments. It is obtained by integrating over all the tjS except for t. ... 

Since one is often interested in deriving the above multi-variable and marginal probability density functions (p.d.f.) from models, it is useful to introduce the multivariable transform function ... 

In heterogeneous systems the theory is not complete but the introduction of multivariable density functions and their transforms and of the associated marginal probability densities seems promising in generalization of various concepts. The powerful central volume principle results from the theory and allows us to evaluate holdups of various phases. For systems with a single flowing phase and transport perpendicular to main flow direction,... 

Scaled marginal probability densities of the mean curvature... 

Scaled marginal probability density of the Gaussian curvature... 

Here Atn(i j) represents the area of i-th triangles sharing thej-th POI. The factor 1/3 is necessary to avoid overcounting the total area by three times, since there are three POI for each triangle. Similar to Eq. 17, the marginal probability densities of the mean curvature, Ph(H), and the Gaussian curvature, Pk(K) are calculated from P(H,K)... 

Fig. 28 Surface contour representation of the joint probability density, P(H,K), measured for the G morphology of the SIS triblock copolymer. Marginal probability densities, Ph(H) and Pk(K), are also shown. The dashed parabolic curve represents K=lfi...

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).

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... 

The single position process X(t) (or velocity process) characterized by the marginal conditional probability density px (or pv) determined from Eq. (4.160), is not in general Markovian. 

For scalar continuous random variables X and Y with joint probability density f (x, y), marginals and conditionals are refined as... 

Figure 3.5 The functions of the probability densities of the one-electron atoms for Is 2s, 2p 3s, 3p, 3d electrons. The numbers on the right margin of the figure are the n values. The xOy plane coincides with the plane of the diagram.

According to Li et al. (2007), it is possible to transform a computation of reliability parameters, using the joint probability density functions of safety margins of system components. 

In reliability analysis, X denotes a vector of random variables that are the input to a model X has the joint probability density function f(X). The response of the physical model as a function of X is described by means of Limit State Functions (LSF) g(X). For RWO, g(X) corresponds to the stop margin of the aircraft, i.e. the length of the runway minus the operational landing distance (the distance actually needed to stop the aircraft). A runway overrun corresponds to the event g(X) < 0 and its probability can be written as ... 

Bayesian statistics has a single way of dealing with nuisance parameters. Because the joint posterior is a probability density in all dimensions, we can find the marginal densities by integration. Inference about the parameter of interest 0i is based on the marginal posterior g 0i data), which is found by integrating the nuisance parameter 2 out of the joint posterior, a process referred to as marginalization ... 

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