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Modeling Bathtub Failure Rates with Hjorth Distribution

Hjorth distribution is used to model lifetimes in reliability studies. For some choice of parameters, its hazard function has a bathtub shape, which corresponds to higher failure probability during a burn-in period, which then lowers and again raises with continuous use. Usually a piece of electronic equipment has an initially high failure rate due to the randomness in quality variations during its production.

Consider a device with its lifetime modeled by Hjorth's distribution with bathtub-shaped hazard function.

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To avoid early failures, the equipment is operated at stress level during a "burn-in" period. Assuming a bathtub hazard function, find the probability of failure within the first year (warranty period).

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Find the length of the burn-in period, after which the failure probability within the first month is reduced by half.

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The expected lifetime of the equipment after having survived the burn-in period.

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Find the time at which the equipment is most reliable.

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In general, the Hjorth distribution is very flexible and with appropriate choice of parameters, it admits many shapes of hazard function, not only bathtub.

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