A certain component is critical to the operation of an electrical system and must be replaced immediately upon failure. If the mean lifetime of this type of component is 100 hours and its standard deviation is 30 hours, how many of the components must be in stock so that the probability that the system is in continual operation for the next 2000 hours is at least. 95?

Question

Mathematics

Kendall Petty

20 July, 18:31

A certain component is critical to the operation of an electrical system and must be replaced immediately upon failure. If the mean lifetime of this type of component is 100 hours and its standard deviation is 30 hours, how many of the components must be in stock so that the probability that the system is in continual operation for the next 2000 hours is at least. 95?

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Chana Keller · Answer 1

n - - √ 1n ∑n i=1 Xi - μ / σ = ∑n i=1 Xi - nμ / √nσ

So both form are equivalent.

LHS has a better interpretation as the terms inside the fraction are finite, and the fraction converge to 00 by LLN. But CLT tell us that by scaling with n - - √ n, the fraction does not goes to infinity; instead it has a normal distribution which is finite almost surely.