WEBVTT

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Every reliability distribution is the same thing,

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described four ways.

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Watch one population of items,

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read through all four functions at once.

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A thousand identical gearbox output bearings,

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commissioned together, running under the same duty.

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As the sweep advances,

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f of t traces the unconditional rate against the whole starting

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population, F of t climbs as the failed share,

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and R of t falls as the surviving share.

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R of t and F of t account for the whole

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population between them, and they sum to one at every point

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on the clock.

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The hazard rate asks a narrower and more useful question.

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It measures failures not against the original population,

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but against the population that has actually survived to time t.

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Divide the two, moment by moment,

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and you get the hazard rate:

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the risk to a survivor,

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not to the original population.

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Watch it climb even as f of t itself is already falling.

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Fewer items are failing in absolute terms,

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but they are a growing share of an ever thinner surviving population.

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Read the last band carefully.

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Against the original thousand bearings,

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only twenty three point eight per cent fail between twenty thousand

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and twenty five thousand hours.

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But of the three hundred and sixty eight bearings that actually

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reached twenty thousand hours,

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sixty four point seven per cent fail before twenty five thousand.

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Both numbers are correct.

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They are answers to different questions.

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A maintenance decision is never made about the original population.

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It is made about the unit in front of you,

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which has already survived to its current age.

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One equation ties all four together:

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R of t equals exp of minus the integral of h

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of tau d tau.

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It says the entire survival history of a population is fully

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determined by its hazard rate history,

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and nothing else.

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That is the spine the rest of this series is built on.