Determining Distribution using normal/chi-square

  • Thread starter Thread starter cse63146
  • Start date Start date
  • Tags Tags
    Distribution
cse63146
Messages
435
Reaction score
0

Homework Statement



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

Homework Equations





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.
 
Physics news on Phys.org
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 ...
 
Ah, they become an F-distribution

so S ~ F2,2

how would I handle the |X| in part b)?
 
cse63146 said:
how would I handle the |X| in part b)?

|X|=\sqrt{X^2}
 
Thank you.
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
Back
Top