If we have $U$, $V$, and $W$ as i.i.d normal RV with mean $0$ and variance $\sigma^2$, then what are the following expressed as a transformation of a known distribution (if known):
1) $\frac{U}{V + W}$ I don't think this can expressed as a distribution.
2) $\frac{U^2}{V^2}$ Both $U^2$ and $V^2$ are $\chi^2$ and the quotient of $\chi^2$ is an F-distribution. I am just not sure what the degrees of freedom are...
3) $\frac{U^2 + V^2}{V^2}$ $U^2 + V^2$ is simply a $\chi^2$ with the sum of the degrees of freedom of $U$ and $V$. A $\chi^2$ divided by another $\chi^2$ is F. But I am wondering is this can be simiplified to $\frac{U^2}{V^2} + 1$ (are you allowed to do that?) in which case it would be $1 + F$...
4) $\frac{U^2 + V^2}{W^2}$ I think it is the same as the one above... just an F distribution.
Please help me out with my reasoning! Thanks.