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Thread: Should be easy, but I'm not getting it somehow

  1. #1
    Actuary.com - Level I Poster
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    Should be easy, but I'm not getting it somehow

    Hello ... I've got a Jacobian transformation problem that starts with two variables, X1 and X2, each of which need to be put into terms of Y1 and Y2.

    I start with Y1 = X1 + X2 and Y2 = X1/X2.

    Of course, I should end up with expressions for X1 and X2 in terms of Y1 and Y2, but I keep coming up with things that reduce to X1=X1, or something equally useless.

    I recognize that in Jacobian transformations, this should actually be one of the easier phases of the problem; it needs to be finished early and without undue delay. I have a feeling that this solution should be terribly obvious, maybe I've just outsmarted myself with this one.

    Is the solution as easy to get as it appears? I must be missing something fairly obvious. I've been provided with final answers, but no idea of how they were arrived at. Any help at all is greatly appreciated.

  2. #2
    Actuary.com - Level I Poster
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    If it's any help, I have the final answers:

    X1 = Y1(Y2)/(1+Y2) and X2 = (Y1)/(1+Y2)

    The problem is, I don't see how to derive them. I suppose I could just start from X1 = X1 (for example), and then just start replacing one of the X1's with Y1's and Y2's; but each replacement seems to require adding both an X1 and an X2, so that's very tricky. I guess I have to find a way where one X1 is replaced by a complex fraction with X2 in both the numerator and denominator (ultimately allowing X2's to be cancelled out), but that still eludes me.

  3. #3
    Actuary.com - Posting Master
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    Y1 = X1 + X2 and Y2 = X1/X2.

    Then 2nd equation implied that X1 = Y2X2.

    Plug that into first equation to get

    Y1 = Y2X2 + X2 = X2(Y2+1) i.e. X2 = Y1/(Y2+1)

    You know that X1 = Y2X2 = Y2Y1/(Y2+1)

  4. #4
    Actuary.com - Level I Poster
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    Thank You!

    That was thorny, since a Jacobian problem has so much more work to be done after that part's complete.

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