bounds for marginal density

lllll · #1 $Old$ November 22nd, 2008, 06:27 PM

$f(x, \ y) = \left\{ \begin{array}{rcl} e^{-(x+y)} & \mbox{for} & x>0, \ y>0 \\ 0 & \mbox{otherwise} \end{array}\right.$

find the marginal density for x

$f_1(x) = \int_0^y e^{-(x+y)} \ dy$

$f_1(x) = e^{-x}\int_0^y e^{-y} \ dy$

$f_1(x) = -e^{-(x+y)} \bigg{|}^y_0$

$f_1(x) = -e^{-(x+y)} + e^{-2x}$

yet the answer in the back of the book is:

$f_1(x) = e^{-x}$

which would imply that the bounds are $-\infty$ and 0 where we would have:

$f_1(x) ={\color{blue}\int_{-\infty}^0} e^{-(x+y)} \ dy$

awkward · #2 $Old$ November 22nd, 2008, 07:03 PM

Quote:

Originally Posted by lllll $View Post$

$f(x, \ y) = \left\{ \begin{array}{rcl}e^{-(x+y)} & \mbox{for} & x>0, \ y>0 \\ 0 & \mbox{otherwise} \end{array}\right.$

find the marginal density for x

$f_1(x) = \int_0^y e^{-(x+y)} \ dy$

[snip]

No, the marginal density of x is given by

$\int_0^\infty e^{-(x+y)} \, dy$