The probability distribution of a quantity with given mean and variance
Abstract
Supplement 1 to the Guide to the Expression of Uncertainty in Measurement states that if a best estimate and associated standard uncertainty are the only information available regarding a certain quantity, then, according to the principle of maximum entropy, a Gaussian probability distribution with support on the real line should be assigned to that quantity. This recommendation holds true if one does not know the range of the possible values of the quantity. However, usually such a range will also be known. It is shown that in this case a Gaussian distribution supported on (-infinity,infinity) can be used, but only if the standard uncertainty is sufficiently small. Otherwise, the parameters of the distribution have to be computed numerically.