Solution Found!
Let Y1 and Y2 be independent normally distributed random
Chapter 5, Problem 131E(choose chapter or problem)
Let \(Y_{1} \text { and } Y_{2}\) be independent normally distributed random variables with means \(\mu_{1} \text { and } \mu_{2}\). respectively, and variances \(\sigma_{1}^{2}=\sigma_{2}^{2}=\sigma^{2}\)
a Show that \(Y_{1} \text { and } Y_{2}\) have a bivariate normal distribution with \(\rho=0\).
b Consider \(U_{1}=Y_{1}+Y_{2}\) and \(U_{2}=Y_{1}-Y_{2}\). Use the result in Exercise to show that
\(U_{1} \text { and } U_{2}\) have a bivariate normal distribution and that \(U_{1} \text { and } U_{2}\) are independent.
Equation Transcription:
Text Transcription:
Y_1 and Y_2
\mu_1 and \mu_2
\sigma_1^2 =\sigma_2^2= \sigma^2
Y_1 and Y_2
\rho=0
U_1=Y_1+Y_2
U_2=Y_1-Y_2
U_1 and U_2
U_1 and U_2
Questions & Answers
QUESTION:
Let \(Y_{1} \text { and } Y_{2}\) be independent normally distributed random variables with means \(\mu_{1} \text { and } \mu_{2}\). respectively, and variances \(\sigma_{1}^{2}=\sigma_{2}^{2}=\sigma^{2}\)
a Show that \(Y_{1} \text { and } Y_{2}\) have a bivariate normal distribution with \(\rho=0\).
b Consider \(U_{1}=Y_{1}+Y_{2}\) and \(U_{2}=Y_{1}-Y_{2}\). Use the result in Exercise to show that
\(U_{1} \text { and } U_{2}\) have a bivariate normal distribution and that \(U_{1} \text { and } U_{2}\) are independent.
Equation Transcription:
Text Transcription:
Y_1 and Y_2
\mu_1 and \mu_2
\sigma_1^2 =\sigma_2^2= \sigma^2
Y_1 and Y_2
\rho=0
U_1=Y_1+Y_2
U_2=Y_1-Y_2
U_1 and U_2
U_1 and U_2
ANSWER:
Solution:
Step 1 of 3:
Let and
be independent normally distributed random variables with means
and variances