Combining relative errors

1Keenan

Hi,

I'm getting confused in combining relative errors.
I have a relative error in evaluating the energy of some particle which is due to a wrong location of the spectrogram centre. this errors affects the energy and the relative error due to the energy resolution of the system.
Thus I have two relative errors: the one due to the centre location and the one due to the resolution which are actually dependet.

How I have to combine both errors? is it correct to sum them?

K^2

So long as two errors are independent, the total error is square root of sum of the squares of errors.

<X + Y> = <X> + <Y>
<(X+Y)²> = <X²> + <Y²> + 2<XY>

Square of the error is the variance, which is defined Var(X) = <X²>-<X>²

Var(X+Y) = Var(X) + Var(Y) + 2Cov(X,Y)

Where covariance, Cov(X,Y) = <XY>-<X><Y> and is zero for independent X and Y.

1Keenan

You are absolutely right, but the thing is that I don't think the errors are not indipendent: the error in the energy due to the wrong centre location affects the error due to the intrinsic resolution of the system...

K^2

You need to figure out how the two are related and estimate covariance. Assume P(X) is normal. Then figure out what you can say about P(Y|X). Sounds from your description like you might expect it to be a normal distribution whose width depends on X. I'm not entirely sure, though. It's not clear why you think the two are related. Maybe you can explain it in more detail.

At any rate, once you have P(X) and P(Y|X), get covariance, and use it with formulae above.

1Keenan

I don't know... maybe it is me making the thing too complicate... I'm just confused and there is no literature about my problem...