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2)] with 1 À GðtÞ 22 TAXONOMY FOR WEIBULL MODELS being the mirror image of FðtÞ. The model was proposed in Cohen (1973) and called the reflected Weibull distribution. 2)] and its reflection about the vertical axis through the origin. As a result, the model is given by the density function ÀÁ gðtÞ ¼ b 12 jtjðbÀ1Þ expðÀjtjb Þ À1

When all the subpopulations are two-parameter Weibull distributions, then we can assume without loss of generality, b1 b2 Á Á Á bn and when bi ¼ bj , i < j, we assume ai ! aj . Model III(c)-2 This is similar to Model III(c)-1 except that the Fi ðtÞ; i ¼ 1; 2; . . ; are inverse Weibull distributions. Model III(c)-3 This is similar to Model III(c)-2 except that some of the subpopulations are Weibull distributions and the remaining are non-Weibull distributions. 4 Sectional Models A general n-fold sectional model is given by 8 k1 F1 ðtÞ > > < " 2 ðtÞ 1 À k2 F GðtÞ ¼ > ÁÁÁ > : " n ðtÞ 1 À kn F 0 t t1 < t t1 t2 ð2:40Þ t > tnÀ1 where the subpopulations Fi ðtÞ are two- or three-parameter Weibull distributions and the ti ’s (called partition points) are an increasing sequence.

The different possible shapes for the hazard function are as follows:     Type Type Type Type 1: Decreasing 2: Constant 3: Increasing 4k: k reverse modal ðk ! 1 Different shapes for the density function. 47 BASIC CONCEPTS [Note: Type 4 ðk ¼ 1Þ is Bathtub shaped and type 8 ðk ¼ 2Þ is W shape]  Type 4k þ 1: k modal ðk ! 1Þ [Note: Type 5 ðk ¼ 1Þ is unimodal and type 9 ðk ¼ 2Þ is bimodal]  Type 4k þ 2: k modal followed by increasing ðk ! 1Þ  Type 4k þ 3: Decreasing followed by k modal ðk ! 2 shows types 1 to 8.

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Probing high-speed digital designs


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