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:

<br /> \begin{align*}<br /> \Delta y_{F} &amp;= \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} \\<br /> &amp;= \Delta y \sqrt{(y_{1}\Delta f(y_{1}))^2 + (y_{2}\Delta f(y_{2}))^2 + ... + (y_{n}\Delta f(y_{n})])^2}<br /> \end{align*}<br />

Is this valid?