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The last lightbulb to burn out

MediumRandom VariablesExpected Value~6m

Four lightbulbs have independent lifetimes X1,X2,X3,X4∼Exponential(1)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. E[max⁡iXi]\mathbb{E}[\max_i X_i])?

Show hints (2)+
  1. The first of 44 to fail is Exp(4)\text{Exp}(4) (mean 1/41/4); by memorylessness the next is Exp(3)\text{Exp}(3), etc.
  2. Sum the successive means 1/4+1/3+1/2+1=H41/4+1/3+1/2+1 = H_4.

Answer

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2.0833 (± 0.001)

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Asked at: SIG, Two Sigma

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