Answer: Let (Y1, Y2) have joint density function fY1 ,Y2

Chapter 6, Problem 113SE

(choose chapter or problem)

Let \(\left(Y_{1}, Y_{2}\right)\) have joint density function f \(y_{1}, y_{2}(\mathrm{y} 1, \mathrm{y} 2)\) and let \(U_{1}=Y_{1} Y_{2} \text { and } U_{2}=Y_{2}\)

Show that the joint density of \(\left(U_{1}, U_{2}\right)\)is

         

\(\mathrm{f} u_{1,} u_{2}\left(u_{1,} u_{2}\right)=\mathrm{f} y_{1,} y_{2}\left(\frac{u 1}{u 2}, u  

                                                           2\right) \frac{1}{|u 2|}\)

Show that the marginal density function for \(\mathrm{U}_{1}\) is

        

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

If \(\mathrm{Y}_{1} \text { and } Y_{2}\) are independent, show that the marginal density function for \(\mathrm{Y}_{1} \text { and } Y_{2}\) is

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

Equation Transcription:

 

      

Text Transcription:

(Y_1, Y_2)  

 y_1, y_2  (y_1, y_2)    

U_1=Y_1Y_2 and U_2=Y_2

(U_1,U_2)

fu_1,u_2(u_1,u_2)=fy_1,y_2(u_1 over u_2, u_2) 1|u_2|

U_1

fu_1(u_1)=-\infty^\infty fy_1,y_2(u_1over u_2,u_2)1|u_2|du_2

Y_1 and Y_2

U_1

fu_1(u_1)=-\infty^\infty fy_1(u_1 over u_2) fy_2(u_2)1|u_2|du_2