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I come from a physics background where the only error propagation I've dealt with was in the lab using the simple formulas found here.

So if I have some function that I am interested in, $f$, and it depends on variable $y$ with some standard deviation $\sigma_y$, then I can compute $\sigma_f$ using one of the above formulas.

From what I understand though, these formulas depend on the probability law for $y$ being a normal distribution.

However, quite often, this is not the case and the probability law for $y$ may be some arbitrary distribution. So my question is, given that we know that the probability distribution function (or uncertainty) for $y$ is not a normal distribution, what are the ways to propagate this uncertainty to $f(y)$?

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From what I understand though, these formulas depend on the probability law for y being a normal distribution.

This is not true: In the first three formulas, the results follow from properties of variance that apply regardless of distribution. (And the fact that $a,b$ are known constants.) To practically demonstrate this, try generating random samples from any distribution, then calculate the sample variance or standard deviation. Compare this with the variance or standard deviation given in the table, and you'll see that they match:

> x1 = runif(100)
> var(x1)
[1] 0.08541137
> var(2*x1)
[1] 0.3416455
> var(x1) * 4
[1] 0.3416455

The others appear to follow the Taylor series approach detailed above the table. (I verified this for $f = a \log(bA)$, but leave the rest to you.) As noted in the text above the table, how well this works depends not on the distribution, but on how well a linear approximation fits the function:

Error estimates for non-linear functions are biased on account of using a truncated series expansion. The extent of this bias depends on the nature of the function. For example, the bias on the error calculated for log x increases as x increases since the expansion to 1+x is a good approximation only when x is small.

In other words, you may use these formulas with the knowledge that error is not normal; but, how well they work depends on how well the function can be linearly approximated. Other methods of error propagation are found here, linked from the same article.

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These formulas do not depend on normality in any way. They simply relate the variance of $f(y)$ to the variance of $y$. For example, the first three rows of that table are exact and hold for any probability distribution. If you could assume that $y$ was normal then many of these formulas could be improved, e.g. when $f(y) = e^y$ and $y$ is normal then the variance can be computed exactly, as opposed to the approximation in the table.

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Check out "Evaluation of measurement data — Supplement 1 to the Guide to the expression of uncertainty in measurement” — Propagation of distributions using a Monte Carlo method," which is available here http://www.bipm.org/utils/common/documents/jcgm/JCGM_101_2008_E.pdf

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