Engineering system of type k-out-of-n is operational if at least k out of n components are...

Engineering system of type k-out-of-n is operational if at least k out of n components are operational. Otherwise, the system fails. Suppose that a k-out-of-n system consists of n identical and independent elements for which the lifetime has Weibull distribution with parameters r and λ. More precisely, if T is a lifetime of a component, P(T ≥ t) = e−λtr, t ≥ 0. Time t is in units of months, and consequently, rate parameter λ is in units (month)−1. Parameter r is dimensionless. Assume that n = 8,k = 4, r = 3/2 and λ = 1/10. (a) Find the probability that a k-out-of-n system is still operational when checked at time t = 3. (b) At the check up at time t = 3 the system was found operational. What is the probability that at that time exactly 5 components were operational? Hint: For each component the probability of the system working at time t is p = e−0.1 t3/2. The probability that a k-out-of-n system is operational corresponds to the tail probability of binomial distribution: IP(X ≥ k), where X is the number of components working. You can do exact binomial calculations or use binocdf in Octave/MATLAB (or dbinom in R, or scipy.stats.binom.cdf in Python when scipy is imported). Be careful with ≤ and <, because of the discrete nature of binomial distribution. Part (b) is straightforward Bayes formula.

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Colby Messinger answered 10 months ago

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