Let $Y_1,...,Y_n \sim Cauchy(0,1), i,i,d.$
How can I find the distribution of $\bar{Y}$?
I know that $Y_1+Y_2\sim Cauchy(0,2)$ by some integration,
but I don't know how to proceed.
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Doesn't what you know immediately imply $(Y_1+Y_2)/2$ has a Cauchy$(0,1)$ distribution? Thus, you already have the answer for $n=2$. Maybe the same kind of integration method you used will extend to larger $n$. – whuber Oct 09 '17 at 18:44