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Chapter 6: Q.6.61 (page 275)

Consider an urn containing n balls numbered 1,.....,nand suppose that k of them are randomly withdrawn. Let Xiequal 1if ball number iis removed and let Xibe 0 otherwise. Show that X1,......Xn are exchangeable .

Short Answer

Expert verified

The probability does not depend on permutation. Hence the variables are exchangeable.

Step by step solution

01

Variable : 

The numerical value or quantity expressed by an alphabetic letter.

02

Explanation : 

Xi=1,if ball iis removed.

And

Xi=0,if ball is inot removed.

Only k out of these variables assume value Xi=1.

And other assume value Xi=0.

Thus, for choosing more or less than k ones in n integers i1,.....,in,

P(鈭�iXi=ii)=0

Since this event is an impossible event in every permutation.

If we have exactly k ones in n integers i1,......in,

P(鈭�iXi=ii)=1nk

All possible configurations of k chosen balls are equally likely due to symmetry.

And there exists nkof these combinations.

Thus,

It has been proved that the probability does not depend on permutation.

Hence, the variables are exchangeable.

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