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, p_a, p_b) distribution if its probability mass function is

$\mathrm{Pr}(U=u)=\{\begin{array}{cc}{p}_a& \text{if }u=a;\\ p_b& \text{if }u=b\\ \frac{1-p_a-p_b}{b-a-1}& \text{if }a<u<b\\ 0& \text{otherwise}\end{array}$

Fitting procedure

Suppose a random variable $X$ has mean $\mu$ and squared coefficient of variation $c^2$. Let $U$ be a Van Houtum distributed random variable. Then the first two moments of $U$ match the first two moments of $X$ if $a$, $b$, $p_a$ and $p_b$ are chosen such that:

$\begin{array}{cc}a& =\lceil \mu -\frac{1}{2}\lceil \sqrt{1+12c^2{\mu }^2}\rceil \rceil \\ b& =\lfloor \mu +\frac{1}{2}\lceil \sqrt{1+12c^2{\mu }^2}\rceil \rfloor \\ p_b& =\frac{(c^2+1){\mu }^2-A-(a^2-A)(2\mu -a-b)/(a-b)}{a^2+b^2-2A}\\ p_a& =\frac{2\mu -a-b}{a-b}+p_b\\ \text{where }A& =\frac{2a^2+a+2ab-b+2b^2}{6}.\end{array}$

There does not exist a Van Houtum distribution for every combination of $\mu$ and $c^2$. By using the fact that for any real mean $\mu$ the discrete distribution on the integers that has minimal variance is concentrated on the integers $\lfloor \mu \rfloor$ and $\lceil \mu \rceil$, 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

$c^2{\mu }^2\ge (\mu -\lfloor \mu \rfloor )(1+\mu -\lceil \mu \rceil {)}^2+(\mu -\lfloor \mu \rfloor {)}^2(1+\mu -\lceil \mu \rceil ).$