I have a = {a(adsbygoogle = window.adsbygoogle || []).push({}); _{1}, a_{2}, .., a_{1000}}, where this set forms a distribution of photoelectrons (pe) seen by a particular photomultiplier tube (pmt) over 1000 repeated events. I then have N sets of these (N pmts), each containing 1000 pe values which I believe are indeed random and independent. So a, b, c, ... (not enough letters!) where a corresponds to pmt_{1}, b corresponds to pmt_{2}etc.

a then goes into a histogram from which a mean and variance is extracted, this is then done for all N sets. I then define N variables pi = μ_{i}/μ_{T}where μ_{i}= mean from histogram i which in turn corresponds to a, and μ_{T}is the sum of the means (μ_{i}) from all N histograms.

I'm just now confused on how I'd calculate V[p_{i}], i.e. for pmt_{1}given that I know E[ a ] and V[ a ]. So far I've thought that perhaps I can say μ_{T}= μ_{1}+ μ_{2}+ ... + μ_{N}= ((a_{1}+ a_{2}+ ... a_{1000}) + (b_{1}+ b_{2}+ ... b_{1000}) +... )/1000 and then therefore V[μ_{T}] = ((V[a_{1}] + V[a_{2}] + ... V[a_{1000}]) + (V[b_{1}] + V[b_{2}] + ... V[b_{1000}]) +... )/1000. I'm not sure however, that this is correct.. First of all I don't know if the logic is correct, and secondly I'm not sure whether the a_{j}, b_{k}and so on can be treated as independent and random variables (I think they are random and independent, but the means E[ a ], E[ b ] etc are not?)

Hopefully this hasn't been too confusing, and any ideas would be greatly appreciated. Cheers.

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# Error propagation for a sum of means

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