Lifetime vs mean time between failure (MTBF) I'm wondering about the difference between MTBF and lifetime (I've to admit I'm not sure whether the last link is too specific..).

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To make it precise: Let's say there is a single mechanical item, what would MTBF and lifetime tell about it?
 A: This depends on the definition of "lifetime" and the underlying probability distribution.
For the exponential decay you show in the second display, if the survival function is $S(t)=\exp (-\lambda t)$, then the mean survival (MTBF, mean time between failures, for independent events) is at $t_{\text{mean}} = 1/\lambda$. That's one potential definition of "lifetime."
In the first display, however, you use median survival as the definition of "lifetime": the time at which half of individuals have experienced the event (survival probability of 0.5). For an exponential survival model, that median survival time is at $t_{1/2}=\ln 2/\lambda$. That's a different, shorter time for this distribution.
Even in this simple situation with an exponential survival model, which definition of "lifetime" do you want to use?
Things become more complicated with other underlying survival distributions or in non-parametric models of censored survival times, like a Cox model. If the last observation time is a censoring time rather than an event time, you can't calculate a "mean" survival time in the usual way with a non-parametric model; you are forced to use something like a "restricted mean survival time" only considering times up to some specified maximum value. That's one reason why median survival is often used as the "lifetime" measure.
