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Capped exponential payout

EasyRandom VariablesExpected Value~3m

A machine's runtime is exponential with mean 44 hours. A warranty pays out the runtime capped at 66 hours - i.e. it pays min⁡(X,6)\min(X, 6). What is the expected payout E[min⁡(X,6)]E[\min(X,6)], in hours? (to 4 decimals)

Show hints (2)+
  1. Mean 4 ⇒ λ = 1/4; E[min⁡(X,c)]=∫0cP(X>t) dt=(1−e−λc)/λE[\min(X,c)] = \int_0^c P(X>t)\,dt = (1-e^{-\lambda c})/\lambda.
  2. Answer =4(1−e−1.5)= 4(1-e^{-1.5}), strictly below the uncapped mean 4.

Answer

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3.1075 (± 0.001)

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Asked at: Game-Based Aptitude, Options Market-Making

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