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Chapter 8: Q. 8.2 (page 393)

Suppose in Problem 8.14that the variance of the number of automobiles sold weekly is9.

(a)Give a lower bound to the probability that next week鈥檚 sales are between 10and22, inclusively.

(b)Give an upper bound to the probability that next week鈥檚 sales exceed18.

Short Answer

Expert verified

(a)the probability is at least .75

(b)P{X>18}鈮�913=.69.

Step by step solution

01

Given Information.

The variance of the number of automobiles sold weekly is9.

02

part (a) Explanation.

Let the random variable Xrepresents the number of automobiles sold (in a week). It is given that E[X]=16and蟽2=Var(X)=9.

(a)By Chebyshev's inequality,

P{|X-16|鈮�6}鈮�蟽262=936=14.

SinceP{10鈮�X鈮�22}=P{|X-16|鈮�6}we get:

P{10鈮�X鈮�22}=1-P{|X-16|鈮�6}鈮�34=.75.

03

Part (b) Explanation.

(b)Using Corollary 5.1.(textbook) fora=2炉we get:

P{X>16+2铃�=18}鈮�蟽2蟽2+22=99+4=913=.69.

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Most popular questions from this chapter

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?

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