Van Houtum distribution
In probability theory and statistics, the Van Houtum distribution is a discrete probability distribution named after prof. Geert-Jan van Houtum. It can be characterized by saying that all values of a finite set of possible values are equally probable, except for the smallest and largest element of this set. Since the Van Houtum distribution is a generalization of the discrete uniform distribution, i.e. it is uniform except possibly at its boundaries, it is sometimes also referred to as quasi-uniform. It is regularly the case that the only available information concerning some discrete random variable are its first two moments. The Van Houtum distribution can be used to fit a distribution with finite support on these moments. A simple example of the Van Houtum distribution arises when throwing a loaded dice which has been tampered with to land on a 6 twice as often as on a 1. The possible values of the sample space are 1, 2, 3, 4, 5 and 6. Each time the die is thrown, the probability of throwing a 2, 3, 4 or 5 is 1/6; the probability of a 1 is 1/9 and the probability of throwing a 6 is 2/9. Probability mass function
A random variable U has a Van Houtum (a, b, pa, pb) distribution if its probability mass function is

Fitting procedure
Suppose a random variable  has mean  and squared coefficient of variation . Let  be a Van Houtum distributed random variable. Then the first two moments of  match the first two moments of  if , ,  and  are chosen such that:

 

There does not exist a Van Houtum distribution for every combination of  and . By using the fact that for any real mean  the discrete distribution on the integers that has minimal variance is concentrated on the integers  and , it is easy to verify that a Van Houtum distribution (or indeed any discrete distribution on the integers) can only be fitted on the first two moments if

References 

Discrete distributions