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Handbook of Reliability, Availability, Maintainability and Safety in Engineering Design - Part 23

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Handbook of Reliability, Availability, Maintainability and Safety in Engineering Design - Part 23 studies the combination of various methods of designing for reliability, availability, maintainability and safety, as well as the latest techniques in probability and possibility modelling, mathematical algorithmic modelling, evolutionary algorithmic modelling, symbolic logic modelling, artificial intelligence modelling, and object-oriented computer modelling, in a logically structured approach to determining the integrity of engineering design. . | 3.3 Analytic Development of Reliability and Performance in Engineering Design 203 Fig. 3.36 Example exponential probability graph c Determining the Maximum Likelihood Estimation Parameter The parameter of the exponential distribution can also be estimated using the maximum likelihood estimation MLE method. This function is log-likelihood and composed of two summation portions f r t s A ln L Niln Xe-XTi - NtXTt 3.167 i 1 i 1 where F is the number of groups of times-to-failure data points. Ni is the number of times to failure in the ith time-to-failure data group. X is the failure rate parameter unknown a priori only one to be found . T is the time of the ith group of time-to-failure data. S is the number of groups of suspension data points. N is the number of suspensions in the ith group of data points. T is the time of the ith suspension data group. The solution will be found by solving for a parameter X so that d A 0 and d A N - T - tift 3.168 dx dx 1 x J 204 3 Reliability and Performance in Engineering Design where also F is the number of groups of times-to-failure data points. Ni is the number of times to failure in the ith time-to-failure data group. A is the failure rate parameter unknown a priori only one to be found . Ti is the time of the ith group of time-to-failure data. 5 is the number of groups of suspension data points. N is the number of suspensions in the ith group of data points. T is the time of the ith suspension data group. 3.3.3.3 Expansion of the Weibull Distribution Model a Characteristics of the Two-Parameter Weibull Distribution The characteristics of the two-parameter Weibull distribution can be exemplified by examining the two parameters ß and p and the effect they have on the Weibull probability density function reliability function and failure rate function. Changing the value of ß the shape parameter or slope of the Weibull distribution changes the shape of the probability density function p.d.f. as shown in Tables 3.15 to 3.19. In .