Ratio of Two Zero-Mean Normal R.Vs So I know that the ratio of two normal distributions is Cauchy. My question is, what are the resulting parameters of the Cauchy R.V., assuming the normal RVs are centered at zero and independent? I'm assuming location is zero, but what about the scale parameter? Thank you in advance.

Here is some basic R code which will allow you to experiment,
f = function(s1,s2){
x1 = rnorm(1e5, mean = 0, sd = s1)
x2 = rnorm(1e5, mean = 0, sd = s2)
y = x1/x2
L = function(p){-sum(dcauchy(y, location = 0, scale = p, log=T))}
optim(1,L)$par
}

What this code does it use MLE to estimate the scale parameter after you specify two sigma values $s_1,s_2$ for the two normals. It uses $10^5$ simulations. For example, $f(2,3) = 0.66$, and $f(2,1) = 2$.
Therefore, the reasonable guess is that the location parameter is $\gamma = \frac{s_1}{s_2}$.
