Solution Found!
Let Y1, Y2, . . . , Yn be independent random variables
Chapter 5, Problem 130E(choose chapter or problem)
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.