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Math Help - trouble with the bounds of integration

  1. #1
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    trouble with the bounds of integration

    verify that this is a valid jdf.

    f(y1, y2)= 6y1^2 y2 0<=y1<=y2, y1+y2<=2
    0 elsewhere

    I was trying to integrate y1 from 0 to y2
    and y2 from 0 to 2

    also i tried y1 from 0 to y2
    and y2 from 0 to 1
    all times 2

    but still not confident with my answers. any help would be great
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  2. #2
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    Quote Originally Posted by batemanl View Post
    verify that this is a valid jdf.

    f(y1, y2)= 6y1^2 y2 0<=y1<=y2, y1+y2<=2
    0 elsewhere

    I was trying to integrate y1 from 0 to y2
    and y2 from 0 to 2

    also i tried y1 from 0 to y2
    and y2 from 0 to 1
    all times 2

    but still not confident with my answers. any help would be great
    A simple sketch graph will show you that the double integral you need to calculate is

    \int_{y_1 = 0}^{y_1 = 1} \int^{y_2 = - y_1 + 2}_{y_2 = y_1} 6 y_1^2 y_2 \, dy_2 \, dy_1,

    where the integral terminals should be evident (especially in hindsight) from a sketch graph of the region over which the joint pdf is non-zero (the triangle bounded by the vertical y_2-axis and the lines y_2 = y_1 and y_2 = -y_1 + 2. Note that the two lines intersect at (1, 1)).
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