Let Z1 and Z2 be independent standard normal random

Chapter 6, Problem 65E

(choose chapter or problem)

Let \(Z_{1}\) and \(Z_{2}\) be independent standard normal random variables and

\(U_{1}=Z_{1}\) and \(U_{2}=Z_{1}+Z_{2}\).

a Derive the joint density of  \(U_{1}\)  and  \(U_{2}\).

b Use Theorem 5.12 to give \(E\left(U_{1}\right), E\left(U_{2}\right), V\left(U_{1}\right), V\left(U_{2}\right)\) and \(\operatorname{Cov}\left(U_{1}, U_{2}\right)\).

c Are \(U_{1}\) and \(U_{2}\) independent? Why?

d Refer to Section 5.10. Show that \(U_{1}\)  and  \(U_{2}\) have a bivariate normal distribution. Identify all the parameters of the appropriate bivariate normal distribution.

Equation Transcription:

Text Transcription:

Z_1

Z_2

U_1=Z_1

U_2=Z_1+Z_2

U_1

U_2

E(U_1),E(U_2),V(U_1),V(U_2)

Cov(U_1,U_2)

U_1

U_2