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Math Help - iid Normal

  1. #1
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    iid Normal

    If X_1 \;,\; X_2 are iid N (0, \sigma^2) random variables with Y_1 = {X_1}^2+{X_2}^2 \; and\; Y_2 = \dfrac{X_1}{\sqrt{{X_1}^2+{X_2}^2}}. The book shows that Y_1\;and\;Y_2 are independent. But how would I find the pdf of \dfrac{X_1}{X_2}??


    I tried doing Let\; U = \dfrac{X_1}{X_2} \;and\; V = {X_2} \implies X_1=UV

    then h_{X_1}(u,v) = uv \;and\;h_{X_2}(u,v)=v

    thus,  J = \left|\begin{array}{cc}v&u\\0&1\end{array}\right|\  ;=\;v

    so, f(u,v) = \dfrac{1}{\sqrt{2\pi\sigma^2}}e^{\frac{(uv)^2}{2\s  igma^2}} \times  \dfrac{1}{\sqrt{2\pi\sigma^2}}e^{\frac{(v)^2}{2\si  gma^2}} \times v

    is this correct? thanks.
    Last edited by chutiya; November 29th 2010 at 10:12 PM.
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  2. #2
    MHF Contributor harish21's Avatar
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    Since X_1 \mbox{and} X_2 are independent:

    f(x_1,x_2)= \dfrac{1}{2\pi\sigma^2} e^{\frac{{-({x_1}^2+{x_2}^2)}}{2\sigma^2}}

    since  x_1 = uv\;,\; x_2=v\;,\;and\; |J|=v

    f(u,v)= 2f(x_1,x_2) \; |J| = ....

    Note that you need to find the pdf of {X_1}/{X_2} = U. You can do so by finding the marginal pdf of f(u,v)
    Last edited by harish21; November 30th 2010 at 07:37 PM. Reason: typo
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