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.

Van Houtum distribution

Probability mass function
Parameters	$p_{a},p_{b}\in [0,1]{\text{ and }}a,b\in \mathbb {Z} {\text{ with }}a\leq b$
Support	$k\in \{a,a+1,\dots ,b-1,b\}\,$
PMF	${\begin{cases}p_{a}&{\text{if }}u=a;\\p_{b}&{\text{if }}u=b\\{\frac {1-p_{a}-p_{b}}{b-a-1}}&{\text{if }}a<u<b\\0&{\text{otherwise}}\end{cases}}$
CDF	${\begin{cases}0&{\textrm {if}}u<a;\\p_{a}&{\text{if }}u=a\\p_{a}+\lfloor x-a\rfloor {\frac {1-p_{a}-p_{b}}{b-a-1}}&{\text{if }}a<u<b\\1&{\text{if }}u\geq b\end{cases}}$
Mean	$ap_{a}+bp_{b}+(1-p_{a}-p_{b}){\frac {a+b}{2}}$
Mode	N/A
Variance	$\ a^{2}p_{a}+b^{2}p_{b}-{}\$ ${\frac {(a+b)(1-p_{a}-p_{b})+2ap_{a}+2bp_{b}}{4}}$ ${}+{\frac {b(2b-1)(b-1)-a(2a+1)(a+1)}{6}}$
Entropy	$\ -p_{a}\ln(p_{a})-p_{b}\ln(p_{b})-{}\$ $(1-p_{a}-p_{b})\ln \left({\frac {1-p_{a}-p_{b}}{b-a-1}}\right)$
MGF	$e^{ta}p_{a}+e^{t}bp_{b}+{\frac {1-p_{a}-p_{b}}{b-a-1}}{\frac {e^{(a+1)t}-e^{bt}}{e^{t}-1}}$
CF	$e^{ita}p_{a}+e^{itb}p_{b}+{\frac {1-p_{a}-p_{b}}{b-a-1}}{\frac {e^{(a+1)it}-e^{bit}}{e^{it}-1}}$

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

$\Pr(U=u)={\begin{cases}p_{a}&{\text{if }}u=a;\\[8pt]p_{b}&{\text{if }}u=b\\[8pt]{\dfrac {1-p_{a}-p_{b}}{b-a-1}}&{\text{if }}a<u<b\\[8pt]0&{\text{otherwise}}\end{cases}}$

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:

${\begin{aligned}a&=\left\lceil \mu -{\frac {1}{2}}\left\lceil {\sqrt {1+12c^{2}\mu ^{2}}}\right\rceil \right\rceil \\[8pt]b&=\left\lfloor \mu +{\frac {1}{2}}\left\lceil {\sqrt {1+12c^{2}\mu ^{2}}}\right\rceil \right\rfloor \\[8pt]p_{b}&={\frac {(c^{2}+1)\mu ^{2}-A-(a^{2}-A)(2\mu -a-b)/(a-b)}{a^{2}+b^{2}-2A}}\\[8pt]p_{a}&={\frac {2\mu -a-b}{a-b}}+p_{b}\\[12pt]{\text{where }}A&={\frac {2a^{2}+a+2ab-b+2b^{2}}{6}}.\end{aligned}}$

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

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