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I'm familiar with the common calculus approach with partial derivatives to evaluate error propagation in calculations with random variables. However, I'm looking for a way to derive the classic formula with the sum of fractional errors squared:

[tex]{\left(\frac{\Delta Z}{Z}\right)}^2 = {\left(\frac{\Delta X}{X}\right)}^2 + {\left(\frac{\Delta Y}{Y}\right)}^2 [/tex]

for the error propagation in a quotient of random variables X & Y:

[tex]Z = X/Y[/tex]

using only the probability density functions (pdf), given that in my specific situation, X & Y are typical gaussian distributions. I've played around with transformations of pdf and multivariate joint pdf (though in my case X & Y are independent), but didn't reach my goal so far.

Is it possible to formally replace the [tex]\Delta X, \Delta Y, \Delta Z[/tex] of the above equation by the variance of the corresponding gaussian pdf? Or am I oblige to resort to Monte Carlo brute force to work with the pdf from the start?

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# Distribution function approach to error propagation

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