Suppose that $X \sim N(\mu_1,\sigma_1)$ and $Y \sim N(\mu_2,\sigma_2)$ are two independent normal random variables. Define $Z = X/Y$. I noticed that there are some cases where the distribution of $Z$ is close to normal


X = rnorm(1000,10,1)
Y = rnorm(1000,10,0.1)

Z = X/Y



Questions. When is the distribution of $Z$ approximately normal? Are there any relevant references to justify this approximation?


1 Answer 1


The Wikipedia page on ratio distributions, in the section on uncorrelated noncentral normal ratios says that a normal approximation is possible "under certain conditions".

The reference given is to Díaz-Francés & Rubio (2013, Statistical Papers). Somewhat unsurprisingly, the abstract notes that the conditions pertain to the coefficients of variation of the two normals.


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