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Math Help - Independent Variables

  1. #1
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    Jun 2010
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    Independent Variables

    Hi,
    I'm having trouble solving the following problem:
    Suppose you have a unit disk (x^2 + y^2 <= 1), and you pick a point randomly from the disk, then let X be the x coordinate and y be the Y coordinate. Are X and Y independent?

    I'm pretty sure the joint distribution is 1/pi when x^2 + y^2 <= 1

    And I know in order to prove independence you have to multiply the marginal distributions, but my problem is I dont know what I have to integrate over to get the marginal distributions. Any help would be greatly appreciated.

    Thanks
    Last edited by imlost22; June 25th 2010 at 08:33 AM. Reason: Clarified the question
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  2. #2
    Senior Member
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    Oct 2009
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    No, X and Y are not independent. If Y = 0, then X can be in [0, 1], whereas if Y = 1, X = 0 necessarily.

    A quick check on dependence in situations like this is to note that independent random variables always have support that is a Cartesian Product of sets i.e. the possible values of X cannot depend on Y and vice versa.
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