There is a couple of error models (probability distributions) where I do understand the derivation in terms of going from simple processes where you have limited knowledge (uncertainty) to the final random variable. For instance, for n Bernoulli trials you get the binomial distribution for the number of "successes" (events). If we go from n experiments to an interval where we think of performing an infinite number of Bernoulli trials with infinitively small success rate, we get the Poisson model. From there, we can ask what to expect about interval times for a given number of events and we get the exponential distribution. Getting the normal distribution can be explained in another route from simple assumptions about errors, e.g. as Herschel showed. Often we deal with strictly positive variables with somehow proportional mean and variance, like shown here:

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3019185/figure/F5/

The normal error model is clearly not appropriate here. Some authors use the log-normal distribution, but this requires multiplicative contributions of errors. In many cases I do not find a theoretic justification for multiplicative errors.

However, the gamma distribution has useful properties to fit such data and the fits are usually better as compared to the log-normal. Unfortunately, I can't find an explanation or derivation of this distribution that theoretically explains why (or if and in which cases) this should/could be applicable. The only thing I found is the derivation from the exponential distribution to provide expectations for the waiting times until a given number of events has occured. Is there any theoretical line of arguments to follow that this also models, say, the number of molecules in a solution, or, as in the linked example, the number of carbonyl residues in a molecule and things like this?

I know the the gamma-distribution gives the expectation about the parameter lambda of the Poisson distribution fro a given number of events (i.e. gamma is the conjugated of the Poisson). I hoped to manage going from there but I'm stuck.

Maybe I'm wrong with the gamma distribution at all. I just have the feeling that this provides the best opportunity for a theoretical justification why our knowledge (or uncertainty) about positive variables should be described by the gamma distribution. I know that other distributions are also used to model right-skewed data but to my impression all these distributions are just taken for granted and used because they simply fit well. I want to get this step beyond the simple "because is fits well". I want to get the "beacuse this distribution correctly expresses what we can know about the underlying processes", as I have for the binomial, Poisson, exponential and normal distributions.

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