Solved – Gaussian Ratio Distribution: Derivatives wrt underlying $mu$’s and $sigma^2$s

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?

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