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I was considering the statistical lifetimes of various light bulbs at first. However, upon further reading it seems that they tend to be approximately Weibull distributed with a shape parameter $k \approx 4,$ not $k=1$ (the latter being required for them to coincide with the exponential distribution).

A great example would be something like a macroscopic analogue of a radioactive atom, with the memorylessness property at play. I suppose you could technically consider an arrival time (of your bus, the next major earthquake, etc) a lifetime of nothing that turns into something after it "decays", but I'd prefer something that doesn't sound like a crude homage to Sartre.

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It might be hard to find a realistic example of exponentially distributed lifetimes (EDL) in macroscopic systems. An EDL requires each member of the population to have a constant probability of failure for each time interval - and if failure doesn't occur during that interval the member's state is 'reset' to its state at the beginning of the time interval. In effect, the member should have no state memory of how many times it has been 'tested' for failure in the past. But in most macroscopic systems failure occurs after some cumulative damage or ageing process which, combined with some natural variability of quality, results in life expectancy statistics consistent with Weibull distribution (or other non-exponential distributions).

So, for example, if your test population is a set of light-bulbs, the probability of failure per time interval won't remain constant as each bulb is suffering damage or deterioration over time - which acts as a 'memory' of past experience. If you compare new bulbs with ones that have run for years, you wouldn't expect the two sets to have identical future life expectancy. This is unlike atoms - if you create a new tritium atom it will have exactly the same future life expectancy as one that has miraculously survived since the big bang.

But you could imagine a situation in which the lifetime appears exponential. Consider a casino manager who is put in charge of a large number of slot machines which can (very) occasionally deliver enormous jackpots. If the manager has no understanding that the machines are (pseudo-) random in operation, he/she may decide that the best way to maximize profit is to weed out the 'generous' machines for scrap. If that was the case, then the population of operating machines would decay exponentially.

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Good answer from Penguino. Here is an example.

Several items of the same type are for sale in a store with many kinds of merchandise. Customers come by at some average rate. On the average, they have a constant probability of wanting one of the items. The lifetime of an item has an exponential distribution.

Examples more along the lines of a light bulb are found in objects that are used at a constant rate and have a constant probability of being lost or damaged with each use. As in "How many times did the chicken cross the road?"

A set of dishes is often missing a plate or two. Every so many uses, one gets dropped. Socks are notorious for going missing in the laundry.

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