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Showing Chi squared is independent with another variable

  1. Dec 11, 2014 #1
    So I have X1 and X2 are iid standard normal.

    Then I have Y=X1^2+ X2^2



    I'm supposed to find the distribution of Y and Z and then determine if they are independent.

    Clearly Y is chi squared with degrees of freedom 2.

    But I have no idea how to find the distribution of Z. I know there is a shortcut without using the jacobian, like I did with the Chi Squares, but I'm not sure how to do it.

    I know Y and Z are not independent because with some algebra,

    So Y depends on Z and vice versa, therefore they cannot be independent. Is that true?
  2. jcsd
  3. Dec 11, 2014 #2


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    Staff: Mentor

    (I moved this thread to the homework section)

    I could be interesting to go to polar coordinates (in the X1-X2-plane).

    Just finding an equation where Y and Z appear is not sufficient to show a dependence.
    Imagine Y=X1, Z=X2, then Y+Z=X1+X2 but they are clearly independent.
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