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If $X_i \sim \mathcal{N}(0,\sigma^2)$ for $i=1,2, \ldots, 2n$ then what is the distribution of $\dfrac{\sum\limits_{i=1}^{n} X_i^2}{\sum\limits_{i=n+1}^{2n} X_i^2}$ ?

Unfortunately I can't figure out how to start working with this problem. Any help?

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Do you need to find a density? If $X_i$ are independent, then according to Wikipedia the distribution is $F_{n,n}$. –  S.D. Nov 30 '11 at 15:37
    
In other terms, the ratio R is such that R/(1+R) is Beta(n/2,n/2). –  Did Nov 30 '11 at 15:50
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It's a non-normalized F-distribution.

The F-distribution (on rare occasions called the Beta distribution of the second kind) is usually defined as that of the quotient $$ \frac{\sum_{i=1}^n X_i^2/n}{\sum_{i=1}^m Y_i^2/m} $$ where $n$ and $m$ are called the numbers of degrees of freedom in the numerator and the denominator, and the $X$s and $Y$s are independent and distributed as $\mathcal{N}(0,1)$.

"Non-normalized" means only that the divisions by $n$ and $m$ are not done. It just re-scales the thing.

There is an immense literature on these, and they are introduced in any statistics textbook that discusses ANOVA.

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