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Suppose that $X=v'A_1v$ and $Y=v'A_2v$, where $A_i$ are symmetric matrices and $v$ a multivariate normal vector with covariance $V$, are chi squared distributed each with its own degrees of freedom. Then it is known that $X$ and $Y$ are independent iff $VA_1VA_2V=0$. In that fortunate case one can work with $X/Y$ which is $F$ distributed.

Unfortunately, in my case $VA_1VA_2V$ is not zero. How can I get the most of the random variable $X/Y$ ? That is: what is its distribution?

edit: What if $X$ is a chi square variable with $r_1$ degrees of freedom and independent from $Y$, and $Y$ is a linear combination of independent central chi square variables? I have been looking for a Matlab implementation, that is all what I need. Linear combinations of chi square variables have already been implemented, so I thought it should be also the case for the independent ratio.

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if X and Y are chi squared and independent,then X/X+Y has beta distribution

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