# Error propagation with dependent variables

Based on Microdosimetry theory, trying to figure out error propagation for a lot of quantities that are produced from radiation spectra where each channel with $f(y)$ counts has error $\sqrt{f(y)}$.

Now, I have a function called the dose-weighted lineal energy distribution:

$d(y) = \frac{yf(y)}{y_{F}} = \frac{yf(y)}{\int{yf(y)dy}}$

I have calculated the constant $y_F\pm\Delta y_F$ using the measured quantity $f(y)\pm\sqrt{f(y)}$ but how do I find the uncertainty in the $d(y)$ distribution when these quantities are not independent? Any help would be greatly appreciated : )

Note: $\Delta y \approx 0$ so this only concerns $f(y)$ and $y_F$.

• Do you know how to propagate errors in general? If yes, what is your precise question about that/why doesn't it work in this case? Commented Oct 26, 2015 at 16:00
• @ACuriousMind - Thanks for your reply. I have determined the errors in $y_F$ using the general method, yes. However, at this point in my process, this is the first time two quantities have not been independent. Unless my understanding of this word is incorrect, the general method cannot be used in this case because they are dependent, right? Commented Oct 26, 2015 at 16:04
• The simplification doesn't work, but the general method still works. The covariances are non-zero then, however. Commented Oct 26, 2015 at 16:14
• Ah ha, gotcha. I haven't ventured into stuff like this before and my maths is a little fuzzy these days - would you happen to know where to find a more detailed description of this general method? Thanks Commented Oct 26, 2015 at 16:20
• Actually, scrap that. About the covariance. Covariance is "a measure of how much two random variables change together" - although $y_F$ has uncertainty based on $f(y)$, it is a constant hence won't the variance always be 0? Commented Oct 26, 2015 at 16:38

Are you able to take multiple measurements to be able to estimate the correlations/covariance involved?

It's not really clear how exactly your channel counts enter the formula, but the "dirty solution" works everytime:

1. Estimate the covariance matrix OR make a lot of observations of correlated counts
2. Based on the estimated covariance matrix, randomly generate a bunch of correlated sets of counts OR just pick the multiple observations you made
3. Plug these counts into your formula for dependent variable OR plug all the observations into the formula
4. Study the distribution of the results (variance, histogram, etc)

The point is, that if you don't require an analytic solution for the error of dependent variable, you can always do it this way and if you have reliable covariance matrix and a lot of generated OR observed sets of counts, you also obtain whole information about distribution of results, not just variance.

• Thanks for the response - I am unable to take the measurements multiple times due to beam-time restrictions; it's just not viable. The channel counts enter my formula like so e.g. For a lineal energy $y = 12$keV/$\mu$m, the normalised counts $f(y) = 0.05 \pm 0.01$ (uncertainty arbitrary in this case) with my spectrum frequency mean $y_F = 24\pm 2$keV/$\mu$m so my distribution for $y = 12$keV/$\mu$m is $d(12) = \frac{12 \times 0.05}{24} = 0.025$. The error in this value is my problem since $y_F$ uses $f(y)$ in its calculation. Commented Oct 27, 2015 at 0:11
• How do you compute $y_F$ ? Do you measure it by measuring $f(y)$ also for other values of $y$ and then compute the integral, or some other method gives you $y_F$ directly? Do you know the shape of $f()$? Commented Oct 29, 2015 at 12:56