Marginal PDF

The marginal PDF of the age, /a(a), will depend on the fluid dynamics and flow geometry.104 On the other hand, the conditional PDF of

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In almost all reported applications of the flamelet model, f and x have been assumed to be independent so that the joint PDF can be written in terms of the marginal PDFs ... 

Thus, the turbulent-reacting-flow problem can be completely closed by assuming independence between Y and 2, and assuming simple forms for their marginal PDFs. In contrast to the conditional-moment closures discussed in Section 5.8, the presumed PDF method does account for the effect of fluctuations in the reaction-progress variable. However, the independence assumption results in conditional fluctuations that depend on f only through Tmax(f ) The conditional fluctuations thus contain no information about local events in mixture-fraction space (such as ignition or extinction) that are caused by the mixture-fraction dependence of the chemical source term. 

Many times, the density of only one of the random variates in a joint pdf is of interest, while the other variables are considered nuisance variables. The density of one random variable alone is called the marginal pdf and is obtained by integrating out (or averaging out) the... 

Figure 2 Updated marginal PDFs at different simulation levels (locally identifiable case)...

Figure 3.17 shows the contours of the marginal PDF of Ki and A 3. It is not surprising that there is negative correlation on the estimates of these two parameters sinee the equivalent linear stiffness of the system is contributed by the combination of Ki and A 3. In this case, the uncertainty for K3 is significantly larger than that of K since the level of the excitation is mild and the nonlinear cubic term is difficult to be identified. 

In order to have better identification results for the nonlinear term, the spectral density of the excitation of the second set of data is increased to 5 = 0.06 s so that the response is of higher level of nonlinearity. Figure 3.18 shows the contours of the marginal PDFs using... 

Then, the excitation level is further increased and identification is proceeded again with two sets of measurements with 5 0 = 0.02 s and = 0.08 s. Figure 3.19 shows the contours of the marginal PDFs. The posterior uncertainty of K3 is further reduced but its updated value departs further from the actual value and this can be considered as biased. The... 

Figure 4.2 shows the contours of the marginal PDF of the natural frequency S2 and the damping ratio The two elliptical contours enclose the region with probability 0.5 and 0.9,... 

Contours of Marginal PDFs for Gaussian Random Variables... 

Two methods are introduced in this appendix for drawing the contours of marginal PDFs. Method 1. Eigenvalue Problem Method... 

In this study, the Nataf model (Ditlevsen Madsen 1996) was used to generate realizations of the random parameters 0. It requires specification of the marginal PDFs of the random parameters 0 and their correlation coefficients. It is therefore able to reproduce the given first- and second-order statistical moments of random parameters 0. The same three-dimensional three-story reinforced concrete building presented in Section 2.4, but on rigid supports, is considered as application example. Table 1 provides the marginal distributions and their statistical parameters for the material parameters modelled as correlated random variables. Other details on the modelling of the structure and the statistical correlation of the random parameters can be found in Barbato et al. (2006). 

Table I Marginal PDFs of material parameters (statistical parameters for lognormal distribution ...

By definition, the random variables u and Y are independent if their joint-PDF is the product of their so-called marginal PDFs ... 

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