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7) is the standard (one-parameter) exponential distribution. This can also be obtained as a special case of the standard Weibull model with the shape parameter b ¼ 1. 6) is often referred to as the power law transformation that links the Weibull and exponential distributions. 8. Model I(b)-2 The model is obtained using the transformation   ZÀt T ¼ b ln Z a ð2:8Þ MODIFICATION/GENERALIZATION OF WEIBULL DISTRIBUTION 23 and is given by  GðtÞ ¼ 1 À exp Àexp tÀt Z ! À1

The two possible shapes for the density function are as follows:  Monotonically decreasing  Unimodal A parametric study is a characterization of the shapes in the two-dimensional parameter plane. It is easily seen that the shape of the density function depends 51 STANDARD WEIBULL MODEL only on the shape parameter b and the scale parameter has no effect. The results are as follows:  For b 1 the density function is monotonically decreasing (type 1 in Fig. 1).  For b > 1 the density function is unimodal with the mode at tm ¼ a½ðb À 1Þ=bŠ1=b (type 2 in Fig.

As a result, they are either a function of the independent variable (t) or some other variables (such as stress level), or are random variables. These lead to the following three subgroups. Type IV(a) Parameters are functions of the variable t (time-varying parameters if t represents time). ). 5 Parameters are random variables (Bayesian models). Type V Models: Discrete Weibull Models In the standard Weibull model, the variable t is continuous and can assume any value in the interval ½0; 1Þ. As a result, T is a continuos random variable.

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