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Thread: Limits of Integration

  1. #1
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    Limits of Integration

    Joint pdf given as kxy for 0 < x < y < 1.

    Find the value of k.





    I understand the process of finding k - doing the double integral and setting it to 1. What I dont understand is the limits of integration for y.

    I've seen two different limits set, but I still cannot seem to figure out how and why it is done.

    I have seen the integral of x from 0 to 1 and the integral of y from x to 1. I have also seen the integral of x from 0 to 1 and the integral of y from 0 to y. Both give the correct answer of k = 8. My question is how do you go about choosing the limits for the y integral?

    Thanks a lot!
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  2. #2
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    Quote Originally Posted by scothoward View Post

    I have seen the integral of x from 0 to 1 and the integral of y from x to 1. I have also seen the integral of x from 0 to 1 and the integral of y from 0 to y. Both give the correct answer of k = 8.
    Of course, it's just reversing integration order.

    Quote Originally Posted by scothoward View Post

    My question is how do you go about choosing the limits for the y integral?
    You mean how you get new bounds when reversing integration order?
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  3. #3
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    Actually, I just dont understand the concept of how the bounds of y are chosen either as x to 1, or 1 to y

    Now that I think about it a bit more...would x to 1 be because y > x, so the lowest y can go is x?? If that is correct, what is the rationale for 1 to y??

    Thanks again
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  4. #4
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    Well, given $\displaystyle 0\le x\le y\le 1,$ bounds of the outter integral should be integers, that's why $\displaystyle 0\le x\le1;$ as for bounds of the inner integral, it's just $\displaystyle x\le y\le1.$
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