Emilio, I am not sure what you mean with the first part of your question (the CDF thing), but my gutt feeling for an answer to this is "no". The answer to the second part of your question is clearly "no".

In order to develop useful models based on empirical data (providing possible functional insights, helping to understand relationships between variables, for quantification of effects, or predictions) we have to formulate our expectations about such data (or, more precisely, about the deviations of the observed data from the data predicted by the model). These expectations are quantified as probabilities (distributions). Obviousely, our expectations will and do depend on what we know about the kind and the possible deviations. This in turn depends on the kind of data we have.

For Bernoulli data (two possible values) the probability distribution consists of just two values (summing to 1; a Bernoulli distribution). A model predicting the outcome of a Bernoulli experiment can be formulated to give the best "expected" result, that is, a weighted average of the possible outcomes. Unless nothing else specific is known, the most appropriate weight factors are the probabilities (=our expectations about the two possible outcomes). But how shall we decide for reasonable expectations? In the abcense of any other information there is no reason why should we prefer one over the other, so on this grounds of information we should resonably assign the same probability to all possible outcomes. It may then turn out from actually observed data or other sources of information, that a different assignment of probabilities is more resonable/useful. In fact, when data is available, we will chose the probabilities so that the likelihood of the data will be maximal. But due to the kind of data it will always be a Bernoulli distribution.

For the same arguments, results from a series of n such experiments, our expectations (of "k successes in n trials") can be formulated as binomial distribution.

Going further, from this probability model can be derived a probability model about counts, leading to the Poisson distribution.

Even one step further, the Poisson distribution can serve as basis to derive resonable expectations about "concentrations" (gamma distribution) as well as of "time-to-events" (exponential distribution). Having a probability distribution for "concentrations" it is relatively simple to derive a probability distribution for "relative concentrations" (proportions), what turns out to be the beta distribution.

This list is not complete.

However, it seems to be quite difficult to derive a reasonable probability distribution for "measurements" (or, better: "errors") that are not bounded (like concentrations or proportions). As far as I know, Gauss actually did not derive the normal distribution from premises about the errors. Instead he took the "reverse" way. He was sure that the arithmetic mean (average) was a optimum solution maximizing the likelihood of the given data. He then found that the normal distribution is the only possible distribution that will eventually lead to this result (based on this he developed the "normal equaltions" and his "least-squares" solution as a numerically tractable short-cut for the calculation of the actual likelihoods). To my opinion, this is not a very convincing way of "deriving" the normal distribution. Luckily, there are several other derivations available. The best to my opinion is the derivation of Herschel (for 2 values), later generalized to 3 and n values by Maxwell and others. The only two assumptions required are: (1) symmetry (positive and negative errors of the same magnitude are expected with the same probability) and (2) independence (knowing one of the errors won't tell us anything about any other error.

See:

http://www-biba.inrialpes.fr/Jaynes/cc07s.pdf

http://courses.ncssm.edu/math/Talks/PDFS/normal.pdf

http://www.sonoma.edu/users/w/wilsonst/Papers/Normal/default.html

So nothing authorizes anyone to request a particular probability distribution. The relevant question that has to be asked is: what can/do I know about the data(errors), and what reasonable expectations can I fromulate based on this knowledge?

If we don't know anything but that the data is continuous, it is not very unreasonable to use the normal distribution since this distribution makes the least assumptions and has the highest entropy. So the normal distribution is a good (i.e. "save") starting point until we know better. Additionally, the likelihood function approximates the likelihood function from normal errors from all other distributions (following from the central limit theorem). So inferences about the parameters based on the likelihood can always be done resonably well using the normal distribution when enough data is available (maybe except for some really strange circumstances).

This method allows us to calculate resonable likelihoods of the data for any parameter value. Often, we just seek the parameter values giving the maximum likelihood (MLE), like the average of measurements (interestingly, the average is also the MLE for binomial and count data). One step further would be to calculate our expectation about a resonable value for the parameter itself, but this already peeks into Bayesian statistics...

ThanKs, Jochen for your comments. CDF is cummulative distribution function which I believe is the main goal of any univariate distribution. Once obtained I have observed that analysts apply different cutting lines according to their specific research experience to measure the probabilities shown by obtained CDF, and then arrive to conclusions to take their decisions. In my not so humble opinion, Probability Density Functions, PDFs, are not so relevant graphs as CDF, and even less if they correspond to Normal Distributions used as a model referent in many text books, papers and statistics books. I ask my self frequently if it is valid to extrapolate the supposed normal distribution of error to the variable. I think error and variable are two completely different research objects, even if variable is used a referent to measure a model fitness -or error-. May you give me some ideas about it? emilio

Emilio, I think I do not understand your post completely. Two things:

(1) The PDF ist simply the derivative of the CDF. There is no substancial difference in using/presenting one or the other. There are *empirical* distribution functions, the relative (cumulative) frequencies of data, and *probability* distribution functions, expressing our expectations about the possible values a variable may take. If lots of data can be obtained under similar conditions of variability and ignorance, a *good* probability distribution (be it represented as PDF or CDF) should be approximated by the empirical distribution (for n->Inf).

(2) As I tried to explain previousely, the normal distribution is *not* a good all-purpose probability distribution. It is particularily "bad" for small data sets where an assymmetric distribution of deviations from the assumed center are expected (e.g. when we have counts values, concentrations, proportions, life-times [and many other things]). Here, our expectations about these deviations can be better represented by other probability distributions (Poisson, binomial, gamma, beta, exponential, ...).

(2.1) If nothing is known and there is no logical reason to assume assymmetric errors, the information-theory proposes the normal distribution because it is the unbounded, continuous distribution with the highest entropy (i.e. uses the least amount of extra-information or assumptions about the data). If one has lots of data, the shape of the probability distribution used has only a negligible influence on the shape of the likelihood function (-> CLT). Therefore, for lots of data, any distribution will serve the purpose to get a "good-enough" likelihood function (to determine the maximum likelihood and the confidence intervals, and, possibly, posterior predictive intervals). Here it is just convenient to take the normal probability because calculations are simple. Using any other distribution would give numerically very similar results (for n->Inf the results would be identical), but it would be a pain in the neck to actually do the calculations (ok, we do have computers today, but there are still no programs doing this easily; and even if so: one can save a lot of computing time and electricity using the "normal approximation").

(2.2) From the above two paragraphs follows that the normal distribution is rarely (I would say: it is *never*(!)) a representation of any real data. It is always an "emergency solution", a "work-around", unless we have (or can practically use) better knowledge about the underlying processes leading to the observed variability. However, if often works good enough, especially on symmetrically distributed data and for large data sets.

And I think I would agree to your statement that "error" and "variable" are not the same. Again: a *probability* distribution represents our *expectations* about values or errors. This is conceptually very different to the actually observed values or errors that might resemble some more or less strange *frequency* distribution under "constant" conditions of variability.

Thanks Jochen for your views. In points (2) you explain some excesses comitted by traditional statisticians that took normal distributions as a universal recipe, so you mention other parametric ones that have moderated those excesses. About point (1) there is probably an error in your opening definition "(1) The PDF ist simply the derivative of the CDF". Its units do not match with expected ones of frequence over X variable units.

When you derivate a model of Lorenz curve you obtain (L2-L1)/delta P ... where P is population frequence; this is an expression of the average value of variable X for a sector with close upper and lower limits -an average that seldom fall in the middle of delta P-. A new second derivate produces something like (delta X)/ (delta P)^2. The inverse of this value is somewhat close to the PDF, but it is very dificult to express the inverse function P(PDF) to integrate it and reach de CDF, so as Gaussians never obtained it, they just postulated a model and recommended it as a general recipe and their parametric tables. It is easier to use data to build a Lorenz curve set of points, then get a good continuous model for it, and then derivate the model to reach CDF. Gaussian followers always request us -their critics- to make a propposal to superate the situation, so that is why I am suggesting in this comment, about a question that may awake this and other alternatives. Thanks Rochen for your kindness with all the RG participants. emilio