1. ## Ratio Distribution

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

2. Originally Posted by arvind
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.

3. 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.