  Supplement 1 to GUM (GUM-S1) recommends the use of maximum entropy principle
(MaxEnt) in determining the probability distribution of a quantity having
specified properties, e.g., specified central moments. When we only know the
mean value and the variance of a variable, GUM-S1 prescribes a Gaussian
probability distribution for that variable. When further information is
available, in the form of a finite interval in which the variable is known to
lie, we indicate how the distribution for the variable in this case can be
obtained. A Gaussian distribution should only be used in this case when the
standard deviation is small compared to the range of variation (the length of
the interval). In general, when the interval is finite, the parameters of the
distribution should be evaluated numerically, as suggested by I. Lira,
Metrologia, 46 L27 (2009). Here we note that the knowledge of the range of
variation is equivalent to a bias of the distribution toward a flat
distribution in that range, and the principle of minimum Kullback entropy (mKE)
should be used in the derivation of the probability distribution rather than
the MaxEnt, thus leading to an exponential distribution with non Gaussian
features. Furthermore, up to evaluating the distribution negentropy, we
quantify the deviation of mKE distributions from MaxEnt ones and, thus, we
rigorously justify the use of GUM-S1 recommendation also if we have further
information on the range of variation of a quantity, namely, provided that its
standard uncertainty is sufficiently small compared to the range.
