I'm working with two independent normal distributions $X$ and $Y$, with means $\mu_x$ and $\mu_y$ and variances $\sigma^2_x$ and $\sigma^2_y$.
I'm interested in the distribution of their ratio $Z=X/Y$. Neither $X$ nor $Y$ has a mean of zero, so $Z$ is not distributed as a Cauchy.
I need to find the CDF of $Z$, and then take the derivative of the CDF with respect to $\mu_x$, $\mu_y$, $\sigma^2_x$ and $\sigma^2_y$.
Does anyone know a paper where these have already been calculated? Or how to do this myself?
I found the formula for the CDF in a 1969 paper, but taking these derivatives will definitely be a huge pain. Maybe someone has already done it or knows how to do it easily? I mainly need to know the signs of these derivatives.
This paper also contains an analytically simpler approximation if $Y$ is mostly positive. I can't have that restriction. However, maybe the approximation has the same sign as the true derivative even outside the parameter range?