Error propagation with dependent variables

[lachy]

Homework Statement

Based on Microdosimetry theory, trying to figure out error propagation for a lot of quantities that are produced from radiation spectra. I am having trouble finding information on how to calculate and propagate errors when the quantities in my equations are not independent.

Homework Equations

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

The Attempt at a Solution

I had attempted doing this with the simplification method that I did in one of my 3rd year stats classes however I realized that this only applies for independent variables; don't know where to go know.

Thanks :)

 

[mfb]

How do you calculate y_F based on f(y)?

 

[lachy]

Edit: sorry, fixing up latex.

Thanks for responding. Given that each channel has been measured independently of the others, the counts in each channel $f(y)\pm \sqrt{f(y)}$ can be used in the "simplification method". To expand on the definition:

$y_F = \int_{0}^{\infty}yf(y)dy = \Delta y \sum_{i = 1}^{n}y_{i}f(y_{i}) = \Delta y(y_{1}f(y_{1}) + y_{2}f(y_{2}) + ... + y_{n}f(y_{n})$

where $\Delta y$ is the lineal energy channel width, no the error in y - forgive my lack of consistency. Anyway, $\Delta \Delta y \approx 0$ so we don't consider it in the error calculation except as a scaling constant.

Treating each $f(y_{i})\pm \Delta f(y_{i})$ as indepdent variables we get:

$\begin{align*} \Delta y_{F} &= \sqrt{(\frac{\partial}{\partial f(y_{1})}[y_{F}]\Delta f(y_{1}))^2 + (\frac{\partial}{\partial f(y_{2})}[y_{F}]\Delta f(y_{2}))^2 + ... + (\frac{\partial}{\partial f(y_{n})}[y_{F}]\Delta f(y_{n}))^2} \\ &= \Delta y \sqrt{(y_{1}\Delta f(y_{1}))^2 + (y_{2}\Delta f(y_{2}))^2 + ... + (y_{n}\Delta f(y_{n})])^2} \end{align*}$

Is this valid?