Van Houtum distribution

In probability theory and statistics, the Van Houtum distribution is a discrete probability distribution named after prof. Geert-Jan van Houtum.^[1] 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	${\displaystyle p_{a},p_{b}\in [0,1]{\text{ and }}a,b\in \mathbb {Z} {\text{ with }}a\leq b}$
Support	${\displaystyle k\in \{a,a+1,\dots ,b-1,b\}\,}$
PMF	${\displaystyle {\begin{cases}p_{a}&{\text{if }}u=a;\\p_{b}&{\text{if }}u=b\\{\frac {1-p_{a}-p_{b}}{b-a-1}}&{\text{if }}a<u<b\\0&{\text{otherwise}}\end{cases}}}$
CDF	${\displaystyle {\begin{cases}0&{\textrm {if}}u<a;\\p_{a}&{\text{if }}u=a\\p_{a}+\lfloor x-a\rfloor {\frac {1-p_{a}-p_{b}}{b-a-1}}&{\text{if }}a<u<b\\1&{\text{if }}u\geq b\end{cases}}}$
Mean	${\displaystyle ap_{a}+bp_{b}+(1-p_{a}-p_{b}){\frac {a+b}{2}}}$
Mode	N/A
Variance	${\displaystyle \ a^{2}p_{a}+b^{2}p_{b}-{}\ }$ ${\displaystyle {\frac {(a+b)(1-p_{a}-p_{b})+2ap_{a}+2bp_{b}}{4}}}$ ${\displaystyle {}+{\frac {b(2b-1)(b-1)-a(2a+1)(a+1)}{6}}}$
Entropy	${\displaystyle \ -p_{a}\ln(p_{a})-p_{b}\ln(p_{b})-{}\ }$ ${\displaystyle (1-p_{a}-p_{b})\ln \left({\frac {1-p_{a}-p_{b}}{b-a-1}}\right)}$
MGF	${\displaystyle e^{ta}p_{a}+e^{t}bp_{b}+{\frac {1-p_{a}-p_{b}}{b-a-1}}{\frac {e^{(a+1)t}-e^{bt}}{e^{t}-1}}}$
CF	${\displaystyle e^{ita}p_{a}+e^{itb}p_{b}+{\frac {1-p_{a}-p_{b}}{b-a-1}}{\frac {e^{(a+1)it}-e^{bit}}{e^{it}-1}}}$

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

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

Fitting procedure

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

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

There does not exist a Van Houtum distribution for every combination of ${\displaystyle \mu }$ and ${\displaystyle c^{2}}$. By using the fact that for any real mean ${\displaystyle \mu }$ the discrete distribution on the integers that has minimal variance is concentrated on the integers ${\displaystyle \lfloor \mu \rfloor }$ and ${\displaystyle \lceil \mu \rceil }$, 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 ^[3]

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