The random variables Y1 and Y2 are independent, both with

QUESTION:

The random variables \(Y_{1} \text { and } Y_{2}\) are independent, both with density

\(f(y)=\left\{\begin{array}{ll}  \frac{1}{y^{2}}, & 1<y \\  0, & \text { otherwise }  \end{array}\right.\)

Let \(U_{1}=\frac{Y_{1}}{Y_{1}+Y_{2}}\)  and \(U_{2}=Y_{1}+Y_{2}\)
a What is the joint density of \(Y_{1} \text { and } Y_{2}\)
b Show that the joint density of \(U_{1} \text { and } U_{2}\) is given by

\(f_{U_{1}, U_{2}}\left(u_{1}, u_{2}\right)=\left\{\begin{array}{ll}

\frac{1}{u_{1}^{2}\left(1-u_{1}\right)^{2} u_{2}^{3}}, & 1 / u_{1}<u_{2}, 0<u_{1}<1 / 2 \text { and } \\

0, & 1 /\left(1-u_{1}\right)<u_{2}, 1 / 2 \leq u_{1} \leq 1, \\  \text { otherwise }  \end{array}\right.\)

c Sketch the region where \(f_{C_{1}, l_{2}}\left(u_{1}, u_{2}\right)>0\)
d Show that the marginal density of \(U_{1}\) is

\(f_{\mathcal{C}_{1}}\left(u_{1}\right)=\left\{\begin{array}{ll}  \frac{1}{2\left(1-u_{1}\right)^{2}} & 0 \leq u_{1}<1 / 2 \\  \frac{1}{2 u_{1}^{2}}, & 1 / 2 \leq u_{1} \leq 1 \\  0, & \text { otherwise }

\end{array}\right.\)

e Are \(U_{1} \text { and } U_{2}\) are independent? Why or why not?

Equation Transcription:

 {

 { 

 {

Text Transcription:

Y1 and Y2

\(f(y)= \frac 1 y^2, & 1<y   0,  otherwise.

U_{1}=\frac{Y_{1}}{Y_{1}+Y_{2}

U2=Y1+Y2

Y1 and Y2

U1 and U2

\(f_U_1, U_2u_1, u_2= \frac1u_1^21-u_1^2 u_2^3, & 1 / u_1<u_2, 0<u_1<1 / 2  0, & 1 1-u_1<u{2, 1 / 2 \leq u_1 \leq 1, { otherwise

f_c_1,l_2(u_1,u_2)> 0

U_1

f_{\mathcal{C}_1(u_1=1 2 1-u_1^2 & 0 \leq u_{1}<1 / 2 \frac 1 2 u_1^2, & 1 / 2 \leq u_1 \leq 1 0, & \text { otherwise }

U_1 and U_2