Ch 6 - 67E

QUESTION:

Let \((Y_1,Y_2)\) have joint density function \(f_{Y_{1}}, y_{2}\left(y_{1}, y_{2}\right)\) and let \(U_{1}=Y_{1} / Y_{2}\) and \(U_{2}=Y_{2}\).

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

                                   \(f_{u_{1}}, v_{2}\left(u_{1}, u_{2}\right)=f_{Y_{1} y_{2}}\left(u_{1} u_{2}, u_{2}\right)\left|u_{2}\right|\)

b Show that the marginal density function for \(U_{1}\)  is

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

c If  and  are independent, show that the marginal density function for \(U_{1}\)  is

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

Equation Transcription:

Text Transcription:

(Y_1,Y_2)

f_Y_1,y2(y_1,y_2)

U_1=Y_/Y_2

U_2=Y_2

(U_1,U_2)

fu_1,v_2(u_1,u2)=f_Y_1,y_2(u_1 u_2,u_2)|u_2|

f_b_1 (u_1)=-integral_-infinity^infinity  f_Y_1,y_2(u_1u_2,u_2)|u_2|du_2

fU1u1=integral_-infinity^infinity  f_Y_1(u_1 u_2)f_Y_2(u_2)|u_2|du_2