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Proabibility - Random variables independence question

  1. Mar 22, 2012 #1
    1. The problem statement, all variables and given/known data
    Two variables, X and Y have a joint density f(x,y) which is constant (1/∏) in the circular region x2+y2 <= 1 and is zero outside that region
    The question is: Are X and Y independent?

    2. Relevant equations
    Well, I know that for two random variable to be independent, multiplication of their marginal densities must equal their joint denstiy, i.e.
    f(x,y)=fX(x)*fY(y)

    3. The attempt at a solution
    My problem is I am confused about how to select the integration limits. I know how to do it when simple boundaries are given (like x<2 and y>1, etc.) but within a circular region, I just could not figuer out how to do it. Should I, for example, integrate y from -sqrt(1-x2 ) to sqrt(1-x2) or is that a wrong approach? How can I select the integration limits in a circular region? Is it a better approach to convert to polar coordinates first and then integrate?

    Thanks a lot for your help.
     
  2. jcsd
  3. Mar 22, 2012 #2
    To be independent means that if I give you value of x, that doesn't increase your knowledge on value of y. If I tell you that x=1, can you tell me anything about y?
     
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