Suppose that Y1 and Y2 are independent exponentially
Chapter 6, Problem 71E(choose chapter or problem)
Suppose that \(Y_{1} \text { and } Y_{2}\) are independent exponentially distributed random variables, both with mean \(\beta\) and define
\(U_{1}=Y_{1}+Y_{2} \text { and } U_{2}=Y_{1} / Y_{2}\)
a. Shoe that the joint density of (\left(U_{1}, U_{2}\right)\) is
\(f_{U_{1}, U_{2}}\left(u_{1}, u_{2}\right)=\left\{\begin{array}{ll} \frac{1}{\beta^{2}} u_{1}
e^{-u_{1} / \beta} \frac{1}{\left(1+u_{2}\right)^{2}}, & 0<u_{1}, 0<u_{2} \\ 0, & \text
{ otherwise } \end{array}\right.\)
b. Are \(U_{1} \text { and } U_{2}\) are independent? Why?
Equation Transcription:
{
Text Transcription:
Y1 and Y2
\beta
U1=Y1+Y2 and U2=Y1/Y2
(U1, U2)
f_{U_1, U_2(u_1, u_2= \frac1\beta^2 u_1 e^-u_1 \beta} \frac11+u_2^2, & 0<u_1, 0<u_2 0, & otherwise
U_1 and U_2
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