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Math Help - Jacobian transformation

  1. #1
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    Jacobian transformation

    Here is a second one I'm not too sure what to do next :

    Let X_1 and X_2 be independent, with respective probability density functions of the forms f_{X_1}(x_1) = x_1 e^{\frac{-x_1^2}{2}}, x_1>0, and f_{X_2}(x_2) = \pi^{-1} (1-x_2^2)^{\frac{-1}{2}}, |x_2|<1.
    Find the distribution of X_1X_2.

    Here's how it goes :
    Let Y_1 = X_1X_2 and Y_2 = X_1.
    The transformation Y = g(X) is then specified by two functions :
    g_1(t_1,t_2) = t_1t_2
    and
    g_2(t_1,t_2) = t_1
    And the inverse transformation X = g^{-1}(Y) is X_1=Y_2 and X_2=\frac{Y_1}{Y_2} giving
    g^{-1}_1(t_1,t_2) = t_2
    and
    g^{-1}_2(t_1,t_2) = \frac{t_1}{t_2}

    So the Jacobian of the transformation is given by
    det \begin{pmatrix} 0 & 1 \\ \frac{1}{y_2} & \frac{-y_1}{y^2_2}  \end{pmatrix}
    So J(y_1,y_2) =\frac{-1}{y_2}
    Hence using the theorem f_{Y_1,Y_2}(y_1,y_2) = f_{X_1,X_2}(y_2,\frac{y_1}{y_2}) |J(y_1,y_2)|<br />
= y_2e^{\frac{-y^2_2}{2}}\pi^{-1}(1-(\frac{y_1}{y_2})^2)^{-1/2}\frac{1}{y_2} = \frac{y_2 e^{\frac{-y^2_2}{2}}}{\pi \sqrt{(y^2_2 -y_1^2)}}

    In order to find f_{Y_1}(y_1) we need to calculate \int f_{Y_1,Y_2}(y_1,y_2) dy_2 over the range of y_2.

    This is where I get stuck. I'm assuming the range of y_2 is the same as the range of x_1 (so  y_2 > 0).
    But it doesn't seem to be integrable.
    Plus I'm not sure what is meant by find the distribution of XY.
    Does it mean identify the distribution of XY (i.e Normal, Gamma..) ?
    Thanks for the help.
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