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Math Help - How to set limits of integration

  1. #1
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    How to set limits of integration

    To find P{X + Y > 3}, where X & Y are jointly distributed random variables, I set the limits of integration to 0-to-1 for dx and 3-x to 5 for dy. This seems reasonable given

    0 < x < 1
    1 < y < 5

    Now, for another question, where I find P{X + Y < 1}, I am told the limits of integration should be from 0 to 1-x for dy and 0-to-1 for dx.

    I don' see why,
    wirefree
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  2. #2
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    Quote Originally Posted by wirefree View Post
    To find P{X + Y > 3}, where X & Y are jointly distributed random variables, I set the limits of integration to 0-to-1 for dx and 3-x to 5 for dy. This seems reasonable given

    0 < x < 1
    1 < y < 5

    Mr F says: The above correct.

    Now, for another question, where I find P{X + Y < 1}, I am told the limits of integration should be from 0 to 1-x for dy and 0-to-1 for dx.

    Mr F asks: For what values of x and y is the joint pdf of X and Y non-zero? If they're the same as above then Pr(X + Y < 1) is obviously zero ....

    I don' see why,
    wirefree
    ..
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  3. #3
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    CORRECTION:

    For P{X + Y < 1}, I am told the limits of integration should be from 0 to 1-x for dy and 0-to-1 for dx given

    0 < x < 1
    0 < y < 2

    Remain in doubt,
    wirefree
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  4. #4
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    Quote Originally Posted by wirefree View Post
    CORRECTION:

    For P{X + Y < 1}, I am told the limits of integration should be from 0 to 1-x for dy and 0-to-1 for dx given

    0 < x < 1
    0 < y < 2

    Remain in doubt,
    wirefree
    You're integrating the joint pdf of X and Y over the triangular region bounded by the x- and y-axes and the line y = 1 - x.

    NB: X + Y < 1 => Y < 1 - X. The solution to this inequality is the region of the XY-plane that lies below the line .....
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