When the final of four bulbs fails

Question

Four light bulbs have independent lifetimes, each exponentially distributed with mean $1$ . How long, on average, until every bulb has burned out?

Short answer

$25/12$ . By memorylessness, the gaps between failures are exponential with rates $4,3,2,1$ (the minimum of $k$ live bulbs has rate $k$ ). The expected time to the last failure is $\tfrac14+\tfrac13+\tfrac12+1=\tfrac{25}{12}$ .

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Lessons behind this question

Continuous Random Variables and Probability Density Functions
Expectation, Mean, and Variance
Conditional Expectation and Variance Revisited

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