Van Houtum distribution

A random variable U has a Van Houtum (a, b, p_a, p_b) distribution if its probability mass function is

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

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]

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