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Chapter 7: Q.7.37 (page 355)

A die is rolled twice. Let X equal the sum of the outcomes, and let Y equal the first outcome minus the second.

ComputeCov(X,Y).

Short Answer

Expert verified

The value ofCov(X,Y)is0.

Step by step solution

01

Given Information

X=The sum of the outcomes,

Y=The first outcome minus the second

Cov(X,Y)=?

02

Explanation

Let Xdenote equal the sum of the outcome.

LetY denote equal the first outcome minus the second.

Let Z1be the outcome of the first role.

Let Z2be the outcome of the second toll.

Thus, X=Z1+Z2andY=Z1-Z2

03

Explanation

Calculate the covariance of the Cov(X,Y)

Cov(X,Y)=CovZ1+Z2,Z1-Z2

=CovZ1,Z1-CovZ1,Z2+CovZ2,Z1-CovZ2,Z2

=CovZ1,Z1-CovZ2,Z2

=VarZ1-VarZ2Since, VarZ1=VarZ2

=0

Here, VarZ1=VarZ2because, Z1 and Z2are identically distributed.

04

Final Answer

The value ofCov(X,Y)is0.

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Most popular questions from this chapter

Urn 1contains 5white and 6black balls, while urn 2contains 8white and 10black balls. Two balls are randomly selected from urn 1and are put into urn 2. If 3balls are then randomly selected from urn 2, compute the expected number of white balls in the trio.

Hint: LetXi = 1if the i th white ball initially in urn 1is one of the three selected, and let Xi = 0otherwise. Similarly, let Yi = 1if the i the white ball from urn 2is one of the three selected, and let Yi = 0otherwise. The number of white balls in the trio can now be written as∑15Xi+∑18Yi

Cards from an ordinary deck are turned face up one at a time. Compute the expected number of cards that need to be turned face up in order to obtain

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(b) the expected number of distinct birthdays.

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Let X1,X2,…be a sequence of independent and identically distributed continuous random variables. Let N≥2be such that

X1≥X2≥⋯≥XN-1<XN

That is, Nis the point at which the sequence stops decreasing. Show that E[N]=e.

Hint: First find P{N≥n}.
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