Chapter 6, Problem 113SE

a.

To determine

Show that the joint density of $\left(U_1,U_2\right)$ is $f_{U_1,U_2}\left(u_1,u_2\right)=f_{Y_1,Y_2}\left(\frac{u_1}{u_2},u_2\right)\frac{1}{\left|u_2\right|}$.

b.

To determine

Show that the marginal density function for $U_1$ is $f_{U_1}\left(u_1\right)={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{f}_{Y_1,Y_2}\left(\frac{u_1}{u_2},u_2\right)\frac{1}{\left|u_2\right|}d{u}_2}$.

c.

To determine

Show that the marginal density function for $U_1$ is $f_{U_1}\left(u_1\right)={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{f}_{Y_1}\left(\frac{u_1}{u_2}\right)f_{Y_2}\left(u_2\right)\frac{1}{\left|u_2\right|}d{u}_2}$.