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Chapter 8: Q. 8.14 (page 391)

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 0.95?

Short Answer

Expert verified

n鈮�23

Step by step solution

01

Given information

饾渿=100and饾湈=30

We need to find n.

Let pbe the probability they will last for atleast 2000 hours.

02

Fomulation

p=P鈭�Xii=1n鈮�2000鈮�PZ鈮�2000-100n30n

where Z is a standard normal random variable.

03

Calculating n

We know that

P(Z>-1.64)=0.95鈬�2000-100n30n鈮�-1.64鈬�2000-100n鈮�-49.2n

Upon using calculator, we get

n鈮�23.

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Most popular questions from this chapter

In Problem8.7, suppose that it takes a random time, uniformly distributed over(0,.5), to replace a failed bulb. Approximate the probability that all bulbs have failed by time550.

From past experience, a professor knows that the test score taking her final examination is a random variable with a mean of75.

(a)Give an upper bound for the probability that a student鈥檚 test score will exceed85.

(b)Suppose, in addition, that the professor knows that the variance of a student鈥檚 test score is equal25. What can be said about the probability that a student will score between 65and85?

(c)How many students would have to take the examination to ensure a probability of at least .9that the class average would be within 5of75? Do not use the central limit theorem.

Each new book donated to a library must be processed. Suppose that the time it takes to process a book has a mean of 10minutes and a standard deviation of 3minutes. If a librarian has 40 books to process,

(a) approximate the probability that it will take more than 420minutes to process all these books;

(b) approximate the probability that at least 25 books will be processed in the first 240 minutes. What assumptions have you made?

Let f(x)be a continuous function defined for0鈮�x鈮�1. Consider the functions

Bn(x)=鈭�k=0nfknnkxk(1鈭�x)n-k(called Bernstein polynomials) and prove that

limn鈫�鈭�Bn(x)=f(x).

Hint: Let X1,X2,...be independent Bernoulli random variables with meanx. Show that

Bn(x)=Efx1+...+xnn

and then use Theoretical Exercise8.4.

Since it can be shown that the convergence of Bn(x)to f(x)is uniformx, the preceding reasoning provides a probabilistic proof of the famous Weierstrass theorem of analysis, which states that any continuous function on a closed interval can be approximated arbitrarily closely by a polynomial.

It Xis a gamma random variable with parameters(n,1), approximately how large must nbe so thatPXn-1>.01<.01?
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