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Math Help - Really confusing distribution problem: standard normals

  1. #1
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    Really confusing distribution problem: standard normals

    If U is uniform on (0,2pi) and Z, independent of U, is exponential with rate 1, show directly that X and Y definited by:

    X=sqrt(2Z)*cosU
    Y=sqrt(2Z)*sinU
    are independent standard normal random variables.

    I have no idea how to even start this one besides just writing the functions for U and Z....
    Last edited by mr fantastic; April 22nd 2009 at 07:45 PM. Reason: Restored question mostly deleted by OP
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  2. #2
    Moo
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    Hello,
    Quote Originally Posted by zhupolongjoe View Post
    If U is uniform on (0,2pi) and Z, independent of U, is exponential with rate 1, show directly that X and Y definited by:

    X=sqrt(2Z)*cosU
    Y=sqrt(2Z)*sinU
    are independent standard normal random variables.

    I have no idea how to even start this one besides just writing the functions for U and Z....
    See here : Box?Muller transform - Wikipedia, the free encyclopedia

    The derivation[3] is based on the fact that, in a two-dimensional cartesian system where X and Y coordinates are described by two independent and normally distributed random variables, the random variables for R2 and Θ (shown above) in the corresponding polar coordinates are also independent and can be expressed as

    R^2 = -2\cdot\ln U_1\,

    and

    \Theta = 2\pi U_2.\,
    fyi, -ln(uniform distribution over (0,1)) follows an exponential distribution with parameter 1. It can be proved by using this : http://en.wikipedia.org/wiki/Inverse_transform_sampling
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    Thanks, even though it doesn't exactly help since it just presents a more muddled version of what the textbook presents.
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    Moo
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    Quote Originally Posted by zhupolongjoe View Post
    Thanks, even though it doesn't exactly help since it just presents a more muddled version of what the textbook presents.
    Huh ?
    Oh no, I'm sorry I completely misread the question

    I don't really know an "immediate" way... But you can find the joint pdf, and then make a 2-2 transform (as matheagle calls it).
    That is a change of variables (namely, changing into polar coordinates)
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    Okay, I know how to compute the Jacobian, but I am unsure about the joint pdf...I can't figure out exactly what it is I should integrate over....\


    EDIT

    disregard...I solved it, thanks though
    Last edited by zhupolongjoe; April 22nd 2009 at 04:27 PM.
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  6. #6
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    Quote Originally Posted by zhupolongjoe View Post
    Okay, I know how to compute the Jacobian, but I am unsure about the joint pdf...I can't figure out exactly what it is I should integrate over....\


    EDIT

    disregard...I solved it, thanks though
    In that case, you might consider posting your solution so that others might benefit from it (or point out mistakes in it .... )
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  7. #7
    MHF Contributor matheagle's Avatar
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    Quote Originally Posted by Moo View Post
    Huh ?
    Oh no, I'm sorry I completely misread the question

    I don't really know an "immediate" way... But you can find the joint pdf, and then make a 2-2 transform (as matheagle calls it).
    That is a change of variables (namely, changing into polar coordinates)
    Yes, it's named after me.
    Mathbeagle 2-2=0 transformation, lol.
    Eye no I'm in trouble when Moo 'quotes' me
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  8. #8
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    Ok here is the solution I did...

    J(R^2,theta)= det(2x 2y; -y/(x^2+y^2) x/(x^2+y^2)
    =2

    Set R^2=2Z
    f(d,theta)=(1/2)*e^(-d/2)*(1/2pi)
    0<d<infinity and 0<theta<2pi
    So R^2 and theta are independent

    We have theta=U and X1=Rcos theta, Y=Rsin theta
    R^2 is chi-squared with 2 degrees of freedom and sowe have from these things

    X=sqrt(2Z) cos U and Y=sqrt(2Z)sin U and they are independent, as we needed
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