The usual method to get anyway the form of a probability distribution, when only a limited number of its moments are known, is to apply the maximum uncertainty principle using Shannon's definition of information. This is not applicable for any positive definite function, which may not represents a probabilty distribution. In that case we propose another type of statistical approach, to be exemplified thinking of an histogram on a finite interval of which only the norm and the second moment are known. Our proposal is to take an average over all good candidates, i. e. the ones with satisfy the imposed bounds, considering all of them equally probable. This is justified by the insufficient reason principle and by the elementary requirement of democracy.

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