My question is: how one derives a Weibull distribution for a lifetime of a system where parts are wearing out with time?

Provided that this is the lifetime analysis with aging is the main application of a Weibull distribution, I was surprised not to find a derivation why specifically this distribution and not any other describe the lifetime of an aging system.

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#### Best Answer

The reason the Weibull distribution is widely used in reliability and life data analysis is most likely due to its versatility. Depending on the parameters used, the Weibull distribution can be used to model a variety of failure laws.

For example, this source http://www.weibull.com/hotwire/issue14/relbasics14.htm provides a good understanding of the versatility of the beta parameter, to quote:

"Effect of beta on Weibull failure rate

This is one of the most important aspects of the effect of β on the Weibull distribution. As is indicated by the plot, Weibull distributions with β < 1 have a failure rate that decreases with time, also known as infantile or early-life failures. Weibull distributions with β close to or equal to 1 have a fairly constant failure rate, indicative of useful life or random failures. Weibull distributions with β > 1 have a failure rate that increases with time, also known as wear-out failures. These comprise the three sections of the classic "bathtub curve." A mixed Weibull distribution with one subpopulation with β < 1, one subpopulation with β = 1 and one subpopulation with β > 1 would have a failure rate plot that was identical to the bathtub curve. An example of a bathtub curve is shown in the following chart."

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