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Chapter 7: Q. 7.9 (page 364)

Nineteen items on the rim of a circle of radius 1are to be chosen. Show that for any choice of these points, there will be an arc of (arc) length 1that contains at least 4of them.

Short Answer

Expert verified

An arc of (arc) length 1that contains at least 4of them isE[X]=192π>3.

Step by step solution

01

Given Information

Nineteen items on the rim of a circle of radius1.

02

Explanation

Let the neighbourhood of any point on the rim be the arc starting at the point and extending for a length 1and let random variable Xdenote the number of points that lie in neighbourhood of chosen point.

Further, let's define indicator variables Ijas:

Ij=10if occurs and does not occur.

03

Explanation

whereby Ej,j=1,2,…,19, denotes the event

Ej="jth item is the neighbourhood of random chosen point ".

X=∑j=119Ij

and therefore the expected number of Xis,

E[X]=E∑j=119Ij=∑j=119EIj=∑j=119PEj=∑j=11912π=192π>3.

04

Final answer

An arc of (arc) length 1that contains at least 4of them isE[X]=192π>3.

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Most popular questions from this chapter

Suppose that X1and X2 are independent random variables having a common mean μ. Suppose also that VarX1=σ12 and VarX2=σ22. The value of μ is unknown, and it is proposed that μ be estimated by a weighted average of X1 and X2. That is, λX1+(1-λ)X2 will be used as an estimate of μ for some appropriate value of λ. Which value of λ yields the estimate having the lowest possible variance? Explain why it is desirable to use this value ofλ.

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