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Question:

  1. Electric light bulbs made at a plant have exponentially distributed lifetime with parameter 2.   We take $1,000$ bulbs.   Estimate the probability that at least one light bulb has life time more than $\tfrac 32(1+\ln 10)$.

My thoughts so far and I get stuck:

this

Lettuce be the lifetime of one light bulb that's more than $\tfrac 32(1+\ln 10)$. $\quad X_k\sim{\exp(2)}$

$\begin{align}\mathsf P( X_k > \tfrac 32(1+\ln 10)) & = \int_{\tfrac 32(I this)}^{\infty} 2 e^{-2t}\operatorname d t \\[1ex] & = \Big[-e^{-2t}\Big]_{\tfrac 32(1+\ln 10)}^{\infty} \\[1ex] &= e^{3(1+\ln 10)} \\[2ex] Y_K &\sim\mathcal{Binom}(1000, e^{3(1+\ln 10)}) \end{align}$

Happy Thanksgiving and thanks in advance to those who helps!!

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  • $\begingroup$ Please learn to post questions (especially ones this brief) and what you have tried instead of just linking to photos. There is a relationship between Poison and exponential distn's. Perhaps define the desired survival as as a Poisson event. $\endgroup$ Commented Nov 24, 2016 at 22:44
  • $\begingroup$ It did not take much work to type it out, except reading the second image was something of a challenge. It's not nice to submit blurry photos of poor handwriting. Please clean it up. $\endgroup$ Commented Nov 25, 2016 at 0:28

1 Answer 1

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You are well under way.

Hint: $\mathsf P(Y_k> 1) = 1-\mathsf P(Y_k=0)$

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