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Handbook of Reliability, Availability, Maintainability and Safety in Engineering Design - Part 25

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Handbook of Reliability, Availability, Maintainability and Safety in Engineering Design - Part 25 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 223 The objective is to interpret the membership function of a fuzzy set as a likelihood function. This idea is not new in fuzzy set theory and has been the basis of experimental design methods for constructing membership functions Loginov 1966 . The likelihood function is a fundamental concept in statistical inference. It indicates how likely a particular set of values will contain an unknown estimated value. For instance suppose an unknown random variable u that has values in the set U is to be estimated. Suppose also that the distribution of u depends on an unknown parameter F with values in the parameter space F. Let P u F be the probability distribution of the variable u where F is the parameter vector of the distribution. If xo is the estimate of variable u an outcome of expert judgment then the likelihood function L is given by the following relationship L F xo P xo F . 3.184 In general both u and xo are vector valued. In other words the estimate xo is substituted instead of the random variable u into the expression for probability of the random variable and the new expression is considered to be a function of the parameter vector F . The likelihood function may vary due to various estimates from the same expert judgment. Thus in considering the probability density function of u at xo denoted by f u F the likelihood functionL is obtained by reversing the roles of F and u that is F is viewed as the variable and u as the estimate which is precisely the point of view in estimation L F u f u F for F inF and u in U. 3.185 The likelihood function itself is not a probability nor density function because its argument is the parameter F of the distribution not the random variable vector u. For example the sum or integral of the likelihood function over all possible values of F should not be equal to 1. Even if the set of all possible values of F is discrete the likelihood function still may