Joint probability curves

When distributions are combined, for example, in joint probability curves, it is important to ensure that the resulting distribution is meaningful, again in terms of what is distributed and with respect to what variable (Suter 1998, p 129). 

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...

Here, H = and K = KL, with Z=0.070 and 0.074 nm for the SIS copolymer and constant-thickness model, respectively. Close examination of the scaled probability densities in Fig. 29a reveals that a part of the scaled joint probability density for the SIS G morphology possesses H < 0 and K>0, implying that the interface is an elliptic surface curved inward relative to the I microphase. Such interfacial concavity is not evident from P(H,K) derived from the constant-thickness model of the G morphology in Fig. 29b, in which nearly all (just under 100%) of the measured points possess k <0. Moreover, P(H,k) of the constant-thickness model exhibits two interesting characteristics. The first is that the measured data are distributed along H = Cok, where the constant Cq is related to the displacement used to construct the constant-thickness model in Fig. 27b from the Schoen G sur-... 

The origin of this unusual behaviour is partly clarified from Fig. 6.34(a) where the relevant curves 2 demonstrate the same kind of the non-monotonous behaviour as the critical exponents above. Since, according to its definition, equation (4.1.19), the reaction rate is a functional of the joint correlation function, this non-monotonicity of curve 2 arises due to the spatial re-arrangements in defect structure. It is confirmed by the correlation functions shown in Fig. 6.34(a). The distribution of BB pairs is quasi-stationary, XB(r,t) X°(r) = exp[(re/r)3], which describes their dynamic aggregation. (The only curve is plotted for XB in Fig. 6.35(a) for t = 102 (the dotted line) since for other time values XB changes not more than by 10 per cent.) This quasi-steady spatial particle distribution is formed quite rapidly already at t 10° it reaches the maximum value of XB(r, t) 103. The effect of the statistical aggregation practically is not observed here, probably, due to the diffusion separation of mobile B particles. 

In the acceptance-sampling simulation, the parameter of interest is 0(p) = Pr F < c p, where Y is the number of defective items in a sample of n items and the quality of the items has a joint Polya distribution with marginal probability p of a defective. As a function of p, 6(p) is the OC curve for sampling plan (n, c). 

Figure C.2 shows the PDF p x, x2 = 7) on the line X2 = 7 and the peak occurs at xi = 11/3, which is the same as the conditional mean. Figure C.3 shows the contours of the joint PDF p x, X2). The two solid lines show the major and minor axes of the elliptical contours. The curves show the contours enclosing the area with 50% and 90% probabilities. The horizontal...

In instrumented creep tests taken to failure, one learns not only how long specimens last but also how deformation increases throughout the creep process. For lap joints, delay times have been seen in creep tests, probably due to the increasing uniformity of the shear stress state, as predicted by the shear lag model as the creep compliance of the adhesive increases with time. In other situations, no such delay time is seen. A schematic illustration of a creep curve for an adhesive bond consisting of a butt joint bonded with a pressure sensitive foam tape is shown in Fig. 2, exhibiting classical primary, secondary and tertiary regions of creep behaviour. 

We close this section by comparing the environmental curve obtained by Monte Carlo estimation to the corresponding contour obtained by the transformation method. Both contours are shown in Figure 13. For this somewhat more extreme joint distribution we observe that the transformation method does not even produce a convex contour. Thus, this contour does not have the required properties. Moreover, most of the contour is biased in the sense that the exceedance probability is underestimated. 

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