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Handbook of Reliability, Availability, Maintainability and Safety in Engineering Design - Part 26

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Handbook of Reliability, Availability, Maintainability and Safety in Engineering Design - Part 26 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 233 Boolean truth table R4 R5 Success or failure Prob. of success Entry R1 R2 R3 1 0 0 0 0 0 F 2 0 0 0 0 1 F 3 0 0 0 1 0 F 4 0 0 0 1 1 F 5 0 0 1 0 0 F 6 0 0 1 0 1 S 0.01008 7 0 0 1 1 0 F 8 0 0 1 1 1 S 0.00252 9 0 1 0 0 0 F 10 0 1 0 0 1 S 0.03888 11 0 1 0 1 0 S 0.03888 12 0 1 0 1 1 S 0.00972 13 0 1 1 0 0 F 14 0 1 1 0 1 S 0.00432 15 0 1 1 1 0 S 0.00432 16 0 1 1 1 1 S 0.00108 17 1 0 0 0 0 F 18 1 0 0 0 1 F 19 1 0 0 1 0 S 0.01008 20 1 0 0 1 1 S 0.00252 etc. g Bayesian Updating Procedure in Reliability Evaluation The elements of a Bayesian reliability evaluation are similar to those for a discrete process considered in Eq. 3.179 above i.e. PCdf . However the structure differs because the failure rate  is well as the reliability RS are continuous-valued. In this case the Bayesian reliability evaluation is given by the formulae p . p P MP- Yi 3203 P i Pi Pi Y n O 3.203 P pl Pi Y where P R IB n V -P Rs P pPYiIRS 3 704x P RslP p ---------pp PY -------- 3.204 234 3 Reliability and Performance in Engineering Design and Aj tC -i . m-k - P RS a f f R 1 - Rs b j number of components with the same A t operating time for determining A and RS a the number of survivals out of j b the number of failures out of j i.e. j a . For both the failure rate A and reliability RS the probability P flj Pj Yj may be either continuous or discrete whereas the probabilities of P Aj for failure and of P RS for reliability are always continuous. Therefore the prior and posterior distributions are always continuous whereas the marginal distribution P flj Pj Yj may be either continuous or discrete. Thus in the case of expert judgment new estimate values in the form of a likelihood function are incorporated into a Bayesian reliability model in a conventional way representing updated information in the form of a posterior a posteriori probability distribution that depends upon a prior a priori probability distribution that in .