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Math Help - Marginal Density

  1. #1
    Junior Member utopiaNow's Avatar
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    Marginal Density

    Given:
    The joint probability density function of X and Y is:
     <br />
f(x, y) = c(y^2 - x^2)e^{-y}<br />

    with bounds  -y < x < y, 0 < y < \infty .

    Question: Find the marginal density of X.
    In the solutions key, when they find the marginal density of X they use the limits of integration |x|\  and\  \infty .

    Why do they do this and not use 0\  and\  \infty as the limits?
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  2. #2
    Moo
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    Hello,
    Quote Originally Posted by utopiaNow View Post
    Given:
    The joint probability density function of X and Y is:
     <br />
f(x, y) = c(y^2 - x^2)e^{-y}<br />

    with bounds  -y < x < y, 0 < y < \infty .

    Question: Find the marginal density of X.
    In the solutions key, when they find the marginal density of X they use the limits of integration |x|\  and\  \infty .

    Why do they do this and not use 0\  and\  \infty as the limits?
    -y<x<y gives :
    . y>x
    . or -y<x ---> y>-x
    This means that y>|x|
    So you have to take \max\{0,|x|\} as the lower bound.
    Since |x|\geq 0, the lower bound will be |x|.

    Is it clearer this way ?
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  3. #3
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    Quote Originally Posted by utopiaNow View Post
    Given:
    The joint probability density function of X and Y is:
     <br />
f(x, y) = c(y^2 - x^2)e^{-y}<br />

    with bounds  -y < x < y, 0 < y < \infty .

    Question: Find the marginal density of X.
    In the solutions key, when they find the marginal density of X they use the limits of integration |x|\ and\ \infty .

    Why do they do this and not use 0\ and\ \infty as the limits?
    Have you tried drawing the region over which the given joint pdf is non-zero? Draw the lines y = x and y = -x. The region is the area above the V-shape, that is the region above the line y = x from x = 0 to x = oo and the region above the line y = -x from x = 0 to x = -oo. This region is alternatively defined by the area above y = |x|.
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  4. #4
    Junior Member utopiaNow's Avatar
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    Thanks guys for the explanations, drawing the region helped me and also understanding that we have to let the lower bound be the max of 0 and |x|.
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