# Mathematical Statistics by Bain Engelhardt

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We are using the book introduction to probability and mathematical statistics by Bain Engelhardt

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6.32 A system is composed of five independent components connected is series.

a) If the pdf of the time to failure of each component is exponential, ,

then give the pdf of the time to failure of the system.

b) Repeat a) but assume that the components are connected in parallel.

c) Suppose that the five-component system fails when at least three components fail.

Give the pdf of the time to failure of the system.

d) Suppose that independent components are not distributed identically, but rather

. Give the pdf of the time to failure of a series system in this case.

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6.32 A system is composed of five independent components connected is series.

a) If the pdf of the time to failure of each component is exponential, ,

then give the pdf of the time to failure of the system.

Solution. Denote by the time to failure of the ith component. By hypothesis, independently.

So,

Note: Since , their identical pdf is

If a system is composed of five independent components connected is series, then the time X to failure of the system is the least time, namely,

So,

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###### Education

- BSc , Wuhan Univ. China
- MA, Shandong Univ.

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- "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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- "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
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