# 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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