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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’s sales are between 10and22, inclusively.

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

Short Answer

Expert verified

(a)the probability is at least .75

(b)P{X>18}≤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]=16andσ2=Var(X)=9.

(a)By Chebyshev's inequality,

P{|X-16|≥6}≤σ262=936=14.

SinceP{10≤X≤22}=P{|X-16|≤6}we get:

P{10≤X≤22}=1-P{|X-16|≥6}≥34=.75.

03

Part (b) Explanation.

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

P{X>16+2⏟=18}≤σ2σ2+22=99+4=913=.69.

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