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A company buys 100 lightbulbs, each of which has an exponential lifetime of 1000 hours. What is the expected time for the first of these bulbs to burn out? For this, I divided 1000/100 to get 10 hours, is this correct?

Bobby
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1 Answers1

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Hint: use that if $X_i\sim$ Exp$(\lambda_i)$ then $\min X_i\sim$ Exp$(\sum\lambda_i)$

You can google the proof, for example here or here

Then you have $\lambda_i=\frac{1}{1000}$, $\sum\lambda_i=\frac{1}{10}$, and the expected lifetime of the first burnt bulb is 10 hours. So your answer is correct.

Momo
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  • How did the 1/1000 go to 1/10? – Bobby Aug 08 '17 at 00:34
  • $E[X_i]=\frac{1}{\lambda_i}=1000$, so $\lambda_i=\frac{1}{1000}$.

    Then $\min X_i \sim$ Exp$(\lambda)$, with $\lambda=\lambda_1+...+\lambda_{100}=\frac{100}{1000}=\frac{1}{10}$.

    Lastly, $E[\min X_i]=\frac{1}{\lambda}=\frac{1}{1/10}=10$

     – Momo Aug 08 '17 at 02:07