System Lifetimes, The Memoryless Property, Euler's Constant, and π

SummaryA k-out-of-n system functions as long as at least k of its n components remain operational. Assuming that component failure times are independent and identically distributed exponential random variables, we find the distribution of system failure time. After some examples, we find the limiting distribution of system life when n approaches infinity, and apply these results to evaluate two challenging integrals.