In a given situation describing a real phenomenon, the random structure of a random variable is defined by its probability distribution (discrete or continuous), the support of the probability distribution being necessarily bounded (i.e. bounded both above and below). The moments of the random variable, which are predetermined as the parameters of the probability distribution, are comparatively easier to estimate than parameters of any other forms. The predetermined moments of the random variable, being the parameters, has to determine the probability distribution uniquely. This is precisely the case, when a given finite number of (arithmetic) moments determines an exponential polynomial probability distribution uniquely. The true probability distribution of a random variable does exist, but is generally unknown and thereby needs to be estimated by certain statistical means. One of such estimators is the minimum information probability distribution, whose type (constant, monotone, uni-extremal or multi-extremal) must be known in advance by means of empirical observations /experiences of the past. A multi-extremal type is not of standard type from the statistical point of view.