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Suppose $x \sim \mathcal N(\mu_x,\sigma_x^2)$ and $y \sim \mathcal N(\mu_y,\sigma_y^2)$ are random variables, and suppose $\mu_y$ is large compared to $\sigma_y$. I want to know about $$ z=\frac{x}{y^2} $$ in particular, its mean and variance.

I'm hoping that since $\mu_y/\sigma_y$ is large (and so unlikely to be 0), this won't be as pathological as the Cauchy distribution. I'm guessing that, in my case, it will be 10 to 100 but it could be larger.

$y^2/\sigma_y^2$ has a noncentral chi-squared distribution with $k=1$ and $\lambda=\mu_y^2/\sigma_y^2$ and so the mean of $y^2$ is $\mu_y^2+\sigma_y^2$ and its variance is $4\mu_y^2+2\sigma_y^2$.

Charles
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    I think $y^2/\sigma_y^2$ has non-central chi-squared distribution with $k = 1$ and $\lambda = \frac{\mu_y^2}{\sigma_y^2}$, but then you would need to calculate the inverse moments of the non-central chi-squared which I believe are in terms of gamma and bessel functions, but not great for closed form computing. – PhysicsKid May 25 '22 at 02:30
  • See https://stats.stackexchange.com/questions/374315/expectation-of-inverse-non-central-chi-squared and https://www.stat.purdue.edu/research/technical_reports/pdfs/1983/tr83-05.pdf – PhysicsKid May 25 '22 at 02:30
  • When $x$ and $y$ are independent, the mean of $z$ is undefined and (therefore) its variance is infinite. Maybe you should consider a different model for $z.$ – whuber May 25 '22 at 13:00
  • @whuber (1) That's disappointing (but I feared it would be the case.) (2) You should add that as an answer. (3) The underlying problem is propagation of standard uncertainties, which are usually given in terms of 1 standard deviation estimates with a normal distribution. If you have other ideas I'd love to hear it -- on a new question, if you like. – Charles May 25 '22 at 13:12
  • Often, Normal distributions work well for modeling errors but they are almost always approximations. Typically the approximation fails in the tails--but rare (left) tail behavior is the cause of the undefined expectation in your case. That's why I suggested reconsidering the model. I can't advance any other ideas without knowing the details. – whuber May 25 '22 at 14:11
  • @whuber Immediate application is surface gravity of an exoplanet based on radius and mass: $g \propto m/r^2$. But error propitiation with division is pretty common, I think, and definitely comes up in a lot of contexts. – Charles May 25 '22 at 14:14
  • A deeper analysis comes down to understanding how the mass and radius were measured and analyzing the statistical characteristics of those measurement systems. – whuber May 25 '22 at 14:19

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