Let Y1, Y2, . . . , Yn be independent random variables

Chapter 5, Problem 130E

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QUESTION:

Let \(Y_{1}, Y_{2}, \ldots, Y_{n}\) be independent random variables with \(E\left(Y_{i}\right)=\mu\) and \(V\left(Y_{i}\right)=\sigma^{2} for i=1,2, \ldots, n\). Let

                                  \(U_{1}=\sum_{i=1}^{n} a_{i} Y_{i} \text { and } U_{2}=\sum_{i=1}^{n} b_{i} Y_{i+}\)

where \(a_{1}, a_{2}, \ldots, a_{n}\), and \(b_{1}, b_{2}, \ldots, b_{n}\) are constants. \(U_{1}\) and \(U_{2}\) are said to be orthogonal if  \(\operatorname{Cov}\left(U_{1}, U_{2}\right)=0\)

a Show that \(U_{1}\) and \(U_{2}\) are orthogonal if and only if \(\sum_{i=1}^{n} a_{i} b_{i}=0\)

b Suppose, in addition, that \(Y_{1}, Y_{2} \ldots\) . \(Y_{n}\) have a multivariate normal distribution. Then \(U_{1}\) and \(U_{2}\) have a bivariate normal distribution. Show that \(U_{1}\) and \(U_{2}\) are independent if they are orthogonal.

Equation Transcription:

Text Transcription:

Y_1,Y_2,...,Y_n

E(Y_i)=mu

V(Y_i)=sigma^2

i=1,2,...,n

U_1=sum over i=1 ^n a_i Y_i and U_2=sum over i=1 ^n b_iY_i

a_i,a_2,...,a_n

b_i,b_2,...,b_n

U_1

U_2

Cov(U_1,U_2)=0

U_1

U_2

Sum over i=1 ^n a_i b_i=0

Y_i,Y_2,...,Y_n

U_1

U_2

U_1

U_2

Questions & Answers

QUESTION:

Let \(Y_{1}, Y_{2}, \ldots, Y_{n}\) be independent random variables with \(E\left(Y_{i}\right)=\mu\) and \(V\left(Y_{i}\right)=\sigma^{2} for i=1,2, \ldots, n\). Let

                                  \(U_{1}=\sum_{i=1}^{n} a_{i} Y_{i} \text { and } U_{2}=\sum_{i=1}^{n} b_{i} Y_{i+}\)

where \(a_{1}, a_{2}, \ldots, a_{n}\), and \(b_{1}, b_{2}, \ldots, b_{n}\) are constants. \(U_{1}\) and \(U_{2}\) are said to be orthogonal if  \(\operatorname{Cov}\left(U_{1}, U_{2}\right)=0\)

a Show that \(U_{1}\) and \(U_{2}\) are orthogonal if and only if \(\sum_{i=1}^{n} a_{i} b_{i}=0\)

b Suppose, in addition, that \(Y_{1}, Y_{2} \ldots\) . \(Y_{n}\) have a multivariate normal distribution. Then \(U_{1}\) and \(U_{2}\) have a bivariate normal distribution. Show that \(U_{1}\) and \(U_{2}\) are independent if they are orthogonal.

Equation Transcription:

Text Transcription:

Y_1,Y_2,...,Y_n

E(Y_i)=mu

V(Y_i)=sigma^2

i=1,2,...,n

U_1=sum over i=1 ^n a_i Y_i and U_2=sum over i=1 ^n b_iY_i

a_i,a_2,...,a_n

b_i,b_2,...,b_n

U_1

U_2

Cov(U_1,U_2)=0

U_1

U_2

Sum over i=1 ^n a_i b_i=0

Y_i,Y_2,...,Y_n

U_1

U_2

U_1

U_2

ANSWER:

Solution:

Step 1 of 3:

Let , ,..., be independent random variables with E() =  and V() =  for  

i = 1,2,...,n

=  and =

Where, , ,..., and , ,..., are constants. and are said to be orthogonal if Cov(,) = 0.


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