The last lightbulb to burn out

MediumRandom Variables Expected Value~6m

Four lightbulbs have independent lifetimes $X_1, X_2, X_3, X_4 \sim \text{Exponential}(1)$ . What is the expected time until the last bulb burns out (i.e. $\mathbb{E}[\max_i X_i]$ )?

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The first of $4$ to fail is $\text{Exp}(4)$ (mean $1/4$ ); by memorylessness the next is $\text{Exp}(3)$ , etc.
Sum the successive means $1/4+1/3+1/2+1 = H_4$ .

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2.0833 (± 0.001)

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The last lightbulb to burn out

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