Ratio Distribution

Mar 2009
3
0
I am trying to find the pdf of the ratio of two RV's \(\displaystyle Z = X/Y\). \(\displaystyle X\) is chi-square distributed with two degrees of freedom and \(\displaystyle Y\) is non-central chi-sqaure distributed with 2 degrees of freedom and non centrality parameter \(\displaystyle \lambda\). How to find the pdf of \(\displaystyle Z\)? Any thoughts?
Another dumb question:
Also, If \(\displaystyle W= X^2/Y^2\), and if \(\displaystyle X\) and \(\displaystyle Y\) are normal distributions with different mean and variances, can I write W as \(\displaystyle (X/Y)^2\) and solve by finding the pdf of \(\displaystyle G = X/Y\) first and then finding the pdf for \(\displaystyle W = G^2\)?

Any suggestion will be really appreciated.

Thanks
Arvind
 

mr fantastic

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Dec 2007
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I am trying to find the pdf of the ratio of two RV's \(\displaystyle Z = X/Y\). \(\displaystyle X\) is chi-square distributed with two degrees of freedom and \(\displaystyle Y\) is non-central chi-sqaure distributed with 2 degrees of freedom and non centrality parameter \(\displaystyle \lambda\). How to find the pdf of \(\displaystyle Z\)? Any thoughts?
Another dumb question:
Also, If \(\displaystyle W= X^2/Y^2\), and if \(\displaystyle X\) and \(\displaystyle Y\) are normal distributions with different mean and variances, can I write W as \(\displaystyle (X/Y)^2\) and solve by finding the pdf of \(\displaystyle G = X/Y\) first and then finding the pdf for \(\displaystyle W = G^2\)?

Any suggestion will be really appreciated.

Thanks
Arvind
In each case, are X and Y independent?

By the way, I suggest you borrow the classic book by M. D. Springer called The Algebra of Random Variables.
 

matheagle

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Feb 2009
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Divide each by 2 and you have an F, BUT there is this noncentrality problem.
So it must be a noncentral F with (2,2) df.