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Thread: Ratio Distribution

  1. #1
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    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
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  2. #2
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    mr fantastic's Avatar
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    Quote Originally Posted by arvind View Post
    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.
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  3. #3
    MHF Contributor matheagle's Avatar
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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.
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