A string of twenty bulbs

Topic: Reliability · more questions on this topic
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Twenty bulbs are wired in series, so the string fails when the first bulb fails. Each bulb’s lifetime is exponential with expectation 1000 hours, and the lifetimes are independent. Find the distribution of the lifetime of the string, and its expectation.

Worked answer (click to show)

The string’s lifetime is T=\min_iT_i. Each T_i has survival function e^{-t/1000}, so

\mathbb{P}(T>t)=\prod_{i=1}^{20}e^{-t/1000}=e^{-20t/1000}=e^{-t/50}.

T is exponential with rate 1/50 per hour, and \mathbb{E} T=50 hours.