Chapter 04 Exercises collection

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Chapter 4 Basics of probability theory for applications to reliability and risk analysis 4.1 Compressors failure Ten compressors, each one with a failure probability of 0.1, are tested independently. 1. What is the expected number of compressors that are found failed? 2. What is the variance of the number of compressors that are found failed? 3. What is the probability that none will fail? 4. What is the probability that two or more will fail? 4.2 Spare unit allocation The allocation of the proper number of spare units for a single­component system is of concern. Assume that if the operating component fails, it is instantaneously replaced by a spare unit, if available. Once the component has failed, it cannot be repaired. The rate of occurrence of components' failures is 1.67y­1, if the component is operating. Assume that the component cannot fail while in the spares depot. The system designer's goal is to achieve a probability of system operation (reliability) of at least 0.95 at the mission time TM of one year. How many spare units should be allocated to achieve this goal? 4.3 Misprints occurrence in a book Consider the occurrence of misprints in a book, and suppose that they occur at the rate of 2 per page. 1. What is the probability that the first misprint will not occur in the first page? 2. Assuming a Poisson process, what is the expected number of pages until the first misprint appears? 3. Comment on the applicability of the Poisson assumptions to the occurrence of misprints or typing errors. 4.4 Peak stresses on a component Suppose that peak stresses (i.e. stresses which exceed a certain value) occur randomly at an average rate Λ. The probability that a component will survive the application of a peak stress is (1­ p) (constant). Show that the reliability of…