Let Y1 and Y2 have joint density function a Derive the

QUESTION:

Let  and  have joint density function

\(f_{Y_{1}, v_{2}}\left(y_{1}, y_{2}\right)=\left\{\begin{array}{ll}  8 y_{1} y_{2}, & 0 \leq y_{1}<y_{2} \leq 1 \\

0, & \text { otherwise }  \end{array}\right.\)

and \(U_{1}=Y_{1} / Y_{2} \text { and } U_{2}=Y_{2}\).

a Derive the joint density function for \(\left(U_{1}, U_{2}\right)\).
b Show that \(U_{1} \text { and } U_{2}\) are independent.

Equation Transcription:

 {

Text Transcription:

Y_1 and Y_2

fy_1,v_2(y1,y2)= { 8y_1y_2,       0\leq  y_1 \leq y_2 \leq 1  0,  otherwise

U_1=Y_1/Y_2 and U_2=Y_2

(U_1,U_2)

U_1 and U_2