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Counts in each bin are Poisson distributed as the counts are random and independent. The background has the same average rate through out the data.

The real world example could be a spot of normally spread long lived radioactive material on ground that is measure by moving a scintillator detector through the spot one meter at a time keeping the measurement time the same in each measurement.

My first idea was to fit the data using non-linear least square method but This, I guess, will fail because of the low number of counts per bin thus the values are not normally distributed but Poisson distributed? Especially a weighted least square method would not work.

Could I use maximum likelihood estimation method to fit a distribution? I would need to get, let's say, 95% or 68% confidence intervals for the fit parameter in addition to the parameter themselves especially for the background.

In practice I am using Matlab for the fitting. http://se.mathworks.com/help/stats/mle.html

An example data:

0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 2, 2, 1, 0, 0, 1, 2, 0, 1, 0, 2, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 0, 2, 2, 1, 3, 10, 15, 5, 6, 2, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 2, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 0, 1, 0

Also, could I use MLE safely for an ash arbitrarily shaped distribution?