The geometric and exponential distributions are said to be memoryless. How much memory do some other distributions require? (Intended as a serious question.)
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Relates to a thing I was wondering about: suppose a process runs for time T before terminating and we know distribution of T. What can we infer about internals of system knowing nothing other than this distribution? If not exponential can we can infer it has some internal state
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Concrete(ish) example: graphics people have started using non-exponential free flight times for photons. Made no sense when I heard it. "Where's the state?" I asked. But in non-homogeneous medium the photon position is state participating in the process benedikt-bitterli.me/nec/
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Another example: a non-exponential failure time for a machine tells you its state is changing, eg. from wear and tear. Exponential suggests (to me at least) wear-and-tear isn't the issue.
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exponential would suggest to me that the system is okay within a few σ from the design optimum and only very rarely and randomly enters a danger/damage zone
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fixed time would mean there's a quantity being used up internally, then the "early or late" failure pattern of hard disks implies a number of bad apples, etc
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Yes. And fixed time implies a lot of memory. Enough to build a clock, because it is, in effect, a clock.
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hmm it could also be a system of a few exponential systems that only breaks down when the % of working systems drops below a threshold
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If we have exponential systems and we wait for first part to fail, we get exponential. If it can survive a % of fails > 0, it's non-exponential. Consistent with memory idea - the set of failed parts comprise the "memory".
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