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Suppose $X_1$ and $X_2$ are two independent random variables that are both $\chi^ 2$-distributed with $n$ and $m$ degrees of freedom, respectively. What is the distribution of $Y=\frac{mX_1}{nX_2}$?

I've tried using the t-student distribution but things got messy real quick...so I suppose there must be a short/elegant solution to this problem?

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