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Math Help - bounds for marginal density

  1. #1
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    bounds for marginal density

    f(x, \ y) = \left\{ \begin{array}{rcl}<br />
e^{-(x+y)} & \mbox{for} & x>0, \ y>0 \\ <br />
0 & \mbox{otherwise} <br />
\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
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  2. #2
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    Quote Originally Posted by lllll View Post
    f(x, \ y) = \left\{ \begin{array}{rcl}<br />
e^{-(x+y)} & \mbox{for} & x>0, \ y>0 \\ <br />
0 & \mbox{otherwise} <br />
\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
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