Showing Chi squared is independent with another variable

  • #1
So I have X1 and X2 are iid standard normal.

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

and

Z=X1/(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,

Y=Y*Z^(2)+X2
So Y depends on Z and vice versa, therefore they cannot be independent. Is that true?
 

Answers and Replies

  • #2
35,268
11,534
(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.
 

Related Threads on Showing Chi squared is independent with another variable

Replies
1
Views
490
Replies
2
Views
672
  • Last Post
Replies
1
Views
4K
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
1
Views
2K
Replies
5
Views
3K
  • Last Post
Replies
2
Views
1K
Replies
3
Views
2K
Top