Determining Distribution using normal/chi-square

  1. 1. The problem statement, all variables and given/known data

    Suppose X,W,Y and Z are all independent. X & Y have a normal distribution. W has N(4,4) distribution while Z has a chi-square with 2 degrees of freedom.

    a) What is the distribution of X2 + Y2 + Z?
    b) What is the distribution of W - 4/(2|X|)
    c) What is the distribution of (X2 + Y2)/Z

    2. Relevant equations



    3. The attempt at a solution

    a) X~N(0,1) Y~N(0,1) Y2~X2(1) X2~X2(1)

    Let S = X2 + Y2 + Z

    therefore S~X2(4)

    c) X~N(0,1) Y~N(0,1) Y2~X2(1) X2~X2(1)

    Let S = (X2 + Y2)/Z

    therefore S~X2(1)

    b) Not sure what to do about the absolute value.
     
  2. jcsd
  3. statdad

    statdad 1,467
    Homework Helper

    Your response to c isn't correct - you cannot say that because the numerator has 2 degrees of freedom, as does the denominator, they cancel.
    You do know that the numerator and denominator are independent, and both have chi-square distributions, so ...
     
  4. Ah, they become an F-distribution

    so S ~ F2,2

    how would I handle the |X| in part b)?
     
  5. [tex]|X|=\sqrt{X^2}[/tex]
     
  6. Thank you.
     
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