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Chapter 7: Q.7.39 (page 355)

Let X1,...be independent with common mean 渭and common variance 蟽2, and set Yn=Xn+Xn+1+Xn+2. For j鈮�0, find CovYn,Yn+j.

Short Answer

Expert verified

The value ofCovYn,Yn+jis0.

Step by step solution

01

Given Information

Independent variable =X1

Mean =渭

Common variance =蟽2

Set functionYn=Xn+Xn+1+Xn+2

For j鈮�0, findCovYn,Yn+j

02

Explanation

From the information, observe that X1,鈥�..be independent with common mean 渭and common variance 蟽2

we have that,

CovYn,Yn=VarXn+Xn+1+Xn+2

=VarXn+VarXn+1+VarXn+2

=蟽2+蟽2+蟽2

=3蟽2

Due to the variance of sum of nindependent variables with common distribution

03

Explanation

If j=1, we have that

CovYn,Yn+1=CovXn+Xn+1+Xn+2,Xn+1+Xn+2+Xn+3

=VarXn+1+VarXn+2

=蟽2+蟽2

=2蟽2

If j=2,we have that,

CovYn,Yn+2=CovXn+Xn+1+Xn+2,Xn+2+Xn+3+Xn+4

=VarXn+2

=蟽2

For j鈮�3,we see that, CovYn,Yn+j=0

Since the definition of Ynand basic properties of covariance to obtain the required.

04

Final Answer

Hence, the value ofCovYn,Yn+jis0.

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