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Math Help - Standard Normal Variable

  1. #1
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    Standard Normal Variable

    Standard Normal Variable-untitled.jpg


    I started by trying to solve for G for x and H for y, so that we could proceed to the Jacobian. However

    it lead to this mess X=cot^-1(2/H) but then I'm left with Y=(H/sin(X))^2/2 (here clearly I don't want an X in the RHS). So I'm stuck. Is there a better way to go about this?
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  2. #2
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    Re: Standard Normal Variable

    Think polar coordinates.  R = \sqrt{X^2 + Y^2} and  \theta = arctan(Y/X)
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  3. #3
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    Re: Standard Normal Variable

    Well you never replied here is the solution
    First notice  g^2 +h^2 = 2y(\cos^2(x) + \sin^2(x)) = 2y giving us the inverse transformation  Y=\frac{G^2 +H^2}{2} .
    Similarily  X= \arctan \left(\frac{H}{G}\right)

     |J| =\left| \begin{array}{cc} g &h \\ \frac{-h}{h^2 + g^2} & \frac{g}{h^2 + g^2} \end{array} \right | = 1

    Thus
     f_{G,H}\left(x=\arctan\left(\frac{h}{g}\right),  y=\frac{g^2 +h^2}{2}\right) |1| = \frac{1}{\sqrt{2 \pi}} e^{-g^2/2} \cdot \frac{1}{\sqrt{2 \pi}} e^{-h^2/2}

    Which are clearly two independent standard normal random variables.
    Last edited by Scopur; December 2nd 2012 at 05:43 PM.
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