Solution
The Gap Between Two Lifetimes
Show the problem again
Two components have independent lifetimes, each exponentially distributed with mean 1 year. What is the expected value of the absolute difference between their lifetimes, in years?
Worked solution
The answer is 1 year. Whichever component fails first, at that moment the other is still running, and by memorylessness its remaining lifetime is a fresh exponential with mean 1, independent of everything so far. The absolute difference |X − Y| is exactly that remaining lifetime, so its expectation is 1. Equivalently, max(X, Y) − min(X, Y): the minimum has mean 1/2 and the maximum has mean 1/2 + 1 = 3/2, and the difference of the means is 1. It is a little surprising that the gap is as large as the mean lifetime itself.
Source: Classic exponential-distribution exercise in wide circulation; no traceable original source. Statement written for AxiomIQ.