Suppose that Y1 and Y2 are independent and that both are
Chapter 6, Problem 70E(choose chapter or problem)
Suppose that \(Y_{1} \text { and } Y_{2}\) are independent and that both are uniformly distributed on the interval, and let \(U_{1}=Y_{1}+Y_{2} \text { and } U_{2}=Y_{1}-Y_{2}\)
a Show that the joint density of \(U_{1} \text { and } U_{2}\) is given by
\(f_{0_{1}, v_{2}}\left(u_{1}, u_{2}\right)=\left\{\begin{array}{ll} 1 / 2, & -u_{1}<u_{2}<u_{1}, 0<u_{1}<1 \text { and } \\ & u_{1}-2<u_{2}<2-u_{1}, 1 \leq u_{1}<2 \\ 0, & \text { otherwise }
\end{array}\right.\)
b Sketch the region where \(f_{v_{1}, v_{2}}\left(u_{1}, u_{2}\right)>0\)
c Show that the marginal density of \(U_{1}\) is
\(f_{0}\left(u_{1}\right)=\left\{\begin{array}{ll} u_{1}, & 0<u_{1}<1 \\ 2-u_{1}, & 1 \leq u_{1}<2 \\
0, & \text { otherwise } \end{array}\right.\)
d Show that the marginal density of \(U_{2}\) is
f_{b_{2}}\left(u_{2}\right)=\left\{\begin{array}{ll} 1+u_{2}, & -1<u_{2}<0 \\
1-u_{2}, & 0 \leq u_{1}<1 \\ 0, & \text { otherwise } \end{array}\right.
e Are \(U_{1} \text { and } U_{2}\) independent? Why or why not?
Equation Transcription:
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Text Transcription:
Y1 and Y2
U1=Y1+Y2 and U2 =Y1-Y2
U1 and U2
\(f_0_1, v_2u_1, u_2\right= 1 / 2, & -u_1<u_2<u_1, 0<u_1<1 and & u_1-2<u_2<2-u_1, 1 \leq u_1<2 0, & otherwise
fv_1,v_2(u_1,u_2)>0
U_1
\(f_0 u_1= u_1, & 0<u_1<1 2-u_1, & 1 \leq u_1<2 0, & otherwise
U_2
F_b_2 u_2=1+u_2, & -1<u_2<0 1-u_2, & 0 \leq u_1<1 0, & otherwise
U_1 and U_2
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