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Chapter 8: Q.8.14 (page 394)

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 these 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?

Short Answer

Expert verified

the number of components of stocks required for given system is

Step by step solution

01

Step 1. Given information

From the given statements of the question, we have to find out 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.

02

Step 2. finding the probability limit

LetXi=lifetimeofithcomponentsgiventhatmeanlifetime渭=EXi=100standarddeviation=30thenvarience(蟽2)=302TakeX=totallifetimeofncomponents,soX=鈭�i=1nXi

Based on the question lifetimes are independent random variables, so by using the properties of expectation we can write,

TherandomvariableXwithmeanEx=E鈭�i=1nXi=鈭�渭i=1n=苍渭=100nVar(x)=Var(鈭�i=1nXi)=鈭�i=1n蟽2=苍蟽2=302蟽2Soaccordingtothequestionwehavetofindpx>2000>0.95

03

Step 3. Finding the number of components using the central limit theorem

The central limit theorem states that , states only that, for each a

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Most popular questions from this chapter

Determine t so that the probability that the repair person in Self-Test Problem 8.7 finishes the 20 jobs within time t is approximately equal to .95.

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