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Math Help - An integration problem

  1. #1
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    An integration problem

    Hi! I'm new here and sure that you guys are able to help me out. I've got stuck with this assignment that I don't know how to solve. My problem is that I don't know what limits I should use for the integrals.

    You can see the problem here:
    http://img103.imagevenue.com/img.php..._122_916lo.jpg

    a) Find P(Y1 < 2, Y2>1)
    I found the right answer to this one, but there are also a B- and a C-assignment, which I don't know how to solve.

    b) Find P(Y1> or equal to 2Y2)

    The correct answer is 1/2.

    c) P(Y1-Y2> or equal to 1)

    The correct answer in this is e^-y1

    Can you please show how to limit the integrals as well?

    Thanks a lot!
    Last edited by mirrormirror; September 19th 2008 at 04:45 AM. Reason: picture is missing
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  2. #2
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    Quote Originally Posted by mirrormirror View Post
    Hi! I'm new here and sure that you guys are able to help me out. I've got stuck with this assignment that I don't know how to solve. My problem is that I don't know what limits I should use for the integrals.

    You can see the problem here:
    http://img103.imagevenue.com/img.php..._122_916lo.jpg

    [snip]
    b) Find P(Y1> or equal to 2Y2)

    The correct answer is 1/2.

    [snip]
    \Pr(Y_1 \geq 2 Y_2) = \int_{y_2 = 0}^{+\infty} \int_{y_1 = 2 y_2}^{+\infty} e^{-y_1} \, dy_1 \, dy_2.
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  3. #3
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    Quote Originally Posted by mirrormirror View Post
    Hi! I'm new here and sure that you guys are able to help me out. I've got stuck with this assignment that I don't know how to solve. My problem is that I don't know what limits I should use for the integrals.

    You can see the problem here:
    http://img103.imagevenue.com/img.php..._122_916lo.jpg

    [snip]

    c) P(Y1-Y2> or equal to 1)

    The correct answer in this is e^-y1 Mr F says: The correct answer is actually e^-1.

    Can you please show how to limit the integrals as well?

    Thanks a lot!
    \Pr(Y_1 - Y_2 \geq 1) = \Pr(Y_1 \geq Y_2 + 1) = \int_{y_2 = 0}^{+\infty} \int_{y_1 = y_2 + 1}^{+\infty} e^{-y_1} \, dy_1 \, dy_2 = e^{-1} = \frac{1}{e}.
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  4. #4
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    Thanks a lot! But why can y2 go from 0 - infinity when there is a restriction that y2 must be smaller than or equal to y1? And can you please show how to calculate the B-assignment as well? I don't get it right anyway. Once again, thanks a lot for your help. And you were right about the answer in C, i read it wrong from the book.
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  5. #5
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    Quote Originally Posted by mirrormirror View Post
    Thanks a lot! But why can y2 go from 0 - infinity when there is a restriction that y2 must be smaller than or equal to y1? Mr F says: The restriction means that the pdf is non-zero only when you're to the right of the Y1 axis and above the line Y1 = Y2 .....

    Draw the region of integration for the question ..... The region is the area to the right of the Y1 axis and above the line {\color{red}y_1 = 2 y_2}. Do you have experience calculating double integrals over a region (it's possible that you don't, but you need to know how to do it for this statistics subject .....)

    And can you please show how to calculate the B-assignment as well? I don't get it right anyway. Once again, thanks a lot for your help. And you were right about the answer in C, i read it wrong from the book.
    \int_{y_2 = 0}^{+\infty} \int_{y_1 = 2 y_2}^{+\infty} e^{-y_1} \, dy_1 \, dy_2 = \int_{y_2 = 0}^{+\infty} \left[ -e^{-y_1} \right]_{2 y_2}^{+\infty} \, dy_2 = \int_{y_2 = 0}^{+\infty} e^{-2 y_2} \, dy_2 = \, ....
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  6. #6
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    Thanks again. I've got two other problems that I've been stuck with for a couple of days. I'd be so glad if someone could help me out with these. Since I'm not very good at writing down the problems, I took a photo of the two assignments that you can see here: http://img121.imagevenue.com/img.php..._122_582lo.jpg

    The problem is the same here, I'm not sure how to limit the integrals, and when I start calculate I always go wrong. So I'd be very grateful for a quite detailed explanation. It'd make my day!
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  7. #7
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    Quote Originally Posted by mirrormirror View Post
    Thanks again. I've got two other problems that I've been stuck with for a couple of days. I'd be so glad if someone could help me out with these. Since I'm not very good at writing down the problems, I took a photo of the two assignments that you can see here: http://img121.imagevenue.com/img.php..._122_582lo.jpg

    The problem is the same here, I'm not sure how to limit the integrals, and when I start calculate I always go wrong. So I'd be very grateful for a quite detailed explanation. It'd make my day!
    New questions should be posted in a new thread.

    For (a) and (b) does the notation mean \Pr\left(-\infty \leq y_1 \leq \frac{1}{2}, ~ -\infty \leq y_2 \leq \frac{1}{2}\right) and \Pr\left(-\infty \leq y_1 \leq \frac{1}{2}, ~ -\infty \leq y_2 \leq 2 \right).

    For all three questions you first need to draw on a set of Y1-Y2 axes the region over which you're doing the double integral. Can you show the region represented by y_1 - 1 \leq y_2 \leq -y_1 + 1 and 0 \leq y_1 \leq 1.

    (a) and (b) Can you show the appropriate part of the above region?

    (c) Can you then show the region corresponding to y_1 > y_2?

    What experience do you have in setting up and solving double integrals? Doing this looks like your real problem - my advice is that you first learn/consolidate/revise this material before trying these sorts of questions.
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  8. #8
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    Thanks for your help. I've managed to solve a and b. And I'm quite good at solving the integrals, my problem is that I don't always know how to limit them. Now I don't know how to limit the integral to solve P(Y1>Y2) I've drawn a picture of the integration area, which you can see here:

    http://img203.imagevenue.com/img.php..._122_494lo.jpg

    The red area is where Y1>Y2, but I need some help to limit the integrals. Thanks in advance.
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  9. #9
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    Quote Originally Posted by mirrormirror View Post
    Thanks for your help. I've managed to solve a and b. And I'm quite good at solving the integrals, my problem is that I don't always know how to limit them. Now I don't know how to limit the integral to solve P(Y1>Y2) I've drawn a picture of the integration area, which you can see here:

    http://img203.imagevenue.com/img.php..._122_494lo.jpg

    The red area is where Y1>Y2, but I need some help to limit the integrals. Thanks in advance.
    The first thing you need to do is get the coordinates of the intersection point of the lines y_2 = y_1 and y_2 = -y_1 + 1. Let them be (a, b). Then one possible double integral is:

    \Pr(Y_1 > Y_2) = \int_{y_2 = y_1 - 1}^{y_2 = y_1} \int_{y_1 = 0}^{y_1 = a} f(y_1, \, y_2) \, dy_2 \, dy_1 + \int_{y_2 = y_1 - 1}^{y_2 = -y_1 + 1} \int_{y_1 = a}^{y_1 = 1} f(y_1, \, y_2) \, dy_2 \, dy_1.


    It's no good being able to integrate if you can't set up the appropriate integrals with the required integral terminals - you obviously have to work on this.
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  10. #10
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    Yeah, I know that I need to practice. Do you know any good internet pages or books that teaches you how to put up double integrals?

    I got the right answer now. Thank you for your help!
    Last edited by mirrormirror; September 27th 2008 at 11:37 PM.
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