Question 2.52: The lifetime of a special type of battery is a random variab......
The lifetime of a special type of battery is a random variable with mean 40 hours and standard deviation 20 hours. A battery is used until it fails, at which point it is replaced by a new one. Assuming a stockpile of 25 such batteries, the lifetimes of which are independent, approximate the probability that over 1100 hours of use can be obtained.
If we let X_{i} denote the lifetime of the ith battery to be put in use, then we desire p=P\{X_{1}+\cdot\cdot\cdot\,+X_{25}\gt \,1100\}, which is approximated as follows:
p=P\left\{{\frac{X_{1}+\cdots+X_{25}-1000}{20{\sqrt{25}}}}\gt{\frac{1100-1000}{20{\sqrt{25}}}}\right\}
≈ P\left\{N(0, 1) > 1\right\}
= 1 − Φ(1)
≈ 0.1587
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