Trisha Shetty (Editor)

Van Houtum distribution

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Parameters
  
p a , p b ∈ [ 0 , 1 ]  and  a , b ∈ Z  with  a ≤ b {\displaystyle p_{a},p_{b}\in [0,1]{\text{ and }}a,b\in \mathbb {Z} {\text{ with }}a\leq b}

Support
  
k ∈ { a , a + 1 , … , b − 1 , b } {\displaystyle k\in \{a,a+1,\dots ,b-1,b\}\,}

pmf
  
{ p a if  u = a ; p b if  u = b 1 − p a − p b b − a − 1 if  a < u < b 0 otherwise {\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
CDF
  
{ 0 if u < a ; p a if  u = a p a + ⌊ x − a ⌋ 1 − p a − p b b − a − 1 if  a < u < b 1 if  u ≥ b {\displaystyle {\begin{cases}0&{\textrm {if}}u
Mean
  
a p a + b p b + ( 1 − p a − p b ) a + b 2 {\displaystyle ap_{a}+bp_{b}+(1-p_{a}-p_{b}){\frac {a+b}{2}}}

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.

Contents

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

Pr ( U = u ) = { p a if  u = a ; p b if  u = b 1 − p a − p b b − a − 1 if  a < u < b 0 otherwise

Fitting procedure

Suppose a random variable X has mean μ 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:

a = ⌈ μ − 1 2 ⌈ 1 + 12 c 2 μ 2 ⌉ ⌉ b = ⌊ μ + 1 2 ⌈ 1 + 12 c 2 μ 2 ⌉ ⌋ p b = ( c 2 + 1 ) μ 2 − A − ( a 2 − A ) ( 2 μ − a − b ) / ( a − b ) a 2 + b 2 − 2 A p a = 2 μ − a − b a − b + p b where  A = 2 a 2 + a + 2 a b − b + 2 b 2 6 .

There does not exist a Van Houtum distribution for every combination of μ and c 2 . By using the fact that for any real mean μ the discrete distribution on the integers that has minimal variance is concentrated on the integers ⌊ μ ⌋ and ⌈ μ ⌉ , 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 μ 2 ≥ ( μ − ⌊ μ ⌋ ) ( 1 + μ − ⌈ μ ⌉ ) 2 + ( μ − ⌊ μ ⌋ ) 2 ( 1 + μ − ⌈ μ ⌉ ) .

References

Van Houtum distribution Wikipedia
(Text) CC BY-SA


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