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Math Help - Figuring out limits of integration when doing joint densities

  1. #1
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    Figuring out limits of integration when doing joint densities

    The problem in question:

    f(x,y)=24xy for 0<= x<=1, 0<=y<=1, 0<=x+y<=1, otherwise x=0

    The goal is to show that f(x,y) is a joint probability fxn and I believe we do that by doubly integrating f(x,y) and showing the result equals 1.

    I believe that the answer is: \int_0^1\int_0^{1-y} \! 24xy \, \mathrm{d}x{d}y, however I'm having trouble understanding the limits of integration. Why is it that we integrate from 0 to 1-y on x and from 0 to 1 on y?

    Here's a similar problem I had earlier:

    f(x,y)=2 0<x<y, 0<y<1

    The problem asks the student to determine whether X,Y are independent. So, I set out to find the marginal distribution of x:

    \int_x^1 \! 2 \, \mathrm{d}y Why do we integrated from x to 1
    \int_0^y \! 2 \, \mathrm{d}x Why do we integrate from 0 to y

    Could we switch the limits of integration as long as we adjust them for both X and Y? For example, could we integrate from 0 to 1 instead of x to 1 if we change the limits of integration on Y?

    Thanks a bunch in advance...I hate probability
    Last edited by crabchef; January 18th 2011 at 01:38 PM. Reason: misplaced [MATH][/MATH] brackets
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  2. #2
    MHF Contributor matheagle's Avatar
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    you need to review calculus three
    To get started you need to draw the regions and that will explain the bounds
    AND by the way I LOVE probability.
    And this seems to be a problem from Walpole, I'm the first reviewer of Walpole in the introduction.
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