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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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A population is made up of r disjoint subgroups. Let pi denote the proportion of the population that is in subgroup i,i=1,鈥�,r. If the average weight of the members of subgroup i is wi,i=1,鈥�,r, what is the average weight of the members of the population?

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