Question

 Let X1, X2 have the joint probability density function f(x1, x2) = 2e −(x1+x2) , 0 < x1, x2 < ∞ Let Y1 = X1, Y2 = X2 − X1.

(i) Using the change of variable technique, find the joint probability density function of Y1, Y2

(ii) Find the conditional distribution of Y2 given Y1

Answer 1

$f\left( x_1,x_2 \right) =2e^{-\left( x_1+x_2 \right)},0<x_1<x_2<\infty \\i:\\\left( Y_1,Y_2 \right) =\left( X_1,X_2-X_1 \right) \\X_1=Y_1\\X_2=Y_1+Y_2\\\left| \frac{\partial \left( X_1,X_2 \right)}{\partial \left( Y_1,Y_2 \right)} \right|=\left| \begin{matrix} 1& 0\\ 1& 1\\\end{matrix} \right|=1\\f_{\left( Y_1,Y_2 \right)}\left( y_1,y_2 \right) =f_{X_1,X_2}\left( y_1,y_1+y_2 \right) \cdot 1=2e^{-\left( y_1+\left( y_1+y_2 \right) \right)}I\left\{ 0<y_1<y_1+y_2<\infty \right\} =\\=2e^{-2y_1}e^{-y_2},y_1>0,y_2>0\\ii:\\Since\\f_{Y_1,Y_2}\left( y_1,y_2 \right) =2e^{-2y_1}I\left\{ y_1>0 \right\} \cdot e^{-y_2}I\left\{ y_2>0 \right\} =f_{Y_1}\left( y_1 \right) f_{Y_2}\left( y_2 \right) ,\\the\,\,conditional\,\,distribution\,\,is\,\,the\,\,distribution\,\,of\,\,Y_2:\\f_{Y_2|Y_1}\left( y_2|y_1 \right) =f_{Y_2}\left( y_2 \right) =e^{-y_2}I\left\{ y_2>0 \right\}$