Let (Y1, Y2) have joint density function fY1 ,Y2 (y1, y2)

Chapter 6, Problem 116SE

(choose chapter or problem)

Let \(\left(Y_{1}, Y_{2}\right)\) have joint density function \(f_{Y_{1}, X_{2}}\left(y_{1} , y_{2}\right)\) and let \(U_{1}=Y_{1}-Y_{2}\) and.

a Show that the joint density of \((U_1,U_2)\) is

\(f_{U_{1}, U_{2}}\left(u_{1}, u_{2}\right)=f_{Y_{1}, Y_{2}}\left(u_{1}+u_{2}, u_{2}\right) .\)

b. Show that the marginal density function for \(U_1\) is

\(f_{U_{1}}\left(u_{1}\right)=\int_{-\infty}^{\infty} f_{Y_{1}, Y_{2}}\left(u_{1}+u_{2}, u_{2}\right) d u_{2}\).

c. If \(Y_1\) and \(Y_2\) are independent, show that the marginal density function for \(U_1\) is

\(f_{U_{1}}\left(u_{1}\right)=\int_{-\infty}^{\infty} f_{Y_{1}}\left(u_{1}+u_{2}\right) f_{Y_{2}}\left(u_{2}\right) d u_{2}\).