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Math Help - is this double integral setup right? f(x,y) = kxy , x>=0, y>= 0 , 20 <= x + y <= 30

  1. #1
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    is this double integral setup right? f(x,y) = kxy , x>=0, y>= 0 , 20 <= x + y <= 30

    So f(x,y) = kxy, it's a joint probability density function and I must find k so that the integral over the valid range is 1.

    My multivariate calculus is a little rusty and my first attempt which looked like (attached) yielded a k of -0.0012 and an integral of 1.

    However I solved for k by double integrating with the calculator and setting the to 1 so of course it will yield 1 for the given interval, what I want to confirm is that the integral was setup properly.

    thanks!

    is this double integral setup right? f(x,y) = kxy , x&gt;=0, y&gt;= 0 , 20 &lt;= x + y &lt;= 30-12-11-2010-7-14-55-pm.jpg
    Last edited by enerj; December 11th 2010 at 04:12 PM.
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  2. #2
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    You shouldn't have an x variable in the inner bounds of integration since you will end up xs when you do the last integration.
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  3. #3
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    OOPS! wrote it wrong in word - the inner integral is dy and outer is dx. Correct then?
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  4. #4
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    With dy and dx switched, there is still a problem.

    If you sketch the region on the plane. It is a trapezoid.
    The bounds for inner integral are different in the regions
    x=0 to x=20
    x=20 to x=30
    because of the y >= 0 condition.
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  5. #5
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    could you propose how to properly setup the double integral? Is the correct region shown below?

    is this double integral setup right? f(x,y) = kxy , x&gt;=0, y&gt;= 0 , 20 &lt;= x + y &lt;= 30-untitled.jpg
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  6. #6
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    Yes, the region is correct.
    What are the bounds for y when x is a certain value?
    This is the same thing as asking, what are the bounds when you take a vertical line slice of a region.

    What are the bounds for y between x = 0 and x = 20?
    What are the bounds for y between x = 20 and x = 30?

    Almost there
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  7. #7
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    so for x between 0 - 20
    y can be [20-x, 30-x]

    for x between 20 - 30
    y can be [0, 30-x]

    how can I show this in the integral?
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  8. #8
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    would it need to look like this?

    is this double integral setup right? f(x,y) = kxy , x&gt;=0, y&gt;= 0 , 20 &lt;= x + y &lt;= 30-12-11-2010-8-00-09-pm.jpg
    and anyone know why only my first attachment is showing itself in the post, and the others need to be clicked on?
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  9. #9
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    Yup, so now split into 2 integrals and add them.

    The first one integrates from x=0 to 20
    and the second integrates from x=20 to 30.

    Yes, exactly what you have in the attachment (I have to click on it for it to open)
    Excellent work.
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