User:BeyondNormality/Aldous distribution

Aldous distribution

Notation	$\mathrm {Aldous} ()$
Parameters	${\displaystyle }$
Support	x ∈ { 3, 4, 5, ... , n}
PMF	${\displaystyle }$
CDF	${\displaystyle }$
Mean	${\displaystyle }$
Median	${\displaystyle }$
Mode	${\displaystyle }$
Variance	${\displaystyle }$
Skewness	${\displaystyle }$
Excess kurtosis	${\displaystyle }$
Entropy	${\displaystyle }$
MGF	${\displaystyle }$
CF	${\displaystyle }$
PGF	${\displaystyle }$

Also known as the distribution of the number of extreme points in the convex hull of a random sample, the probability mass function of the Aldous distribution is given by

${\begin{aligned}\mathrm {Aldous} (x)\equiv \Pr(X=x)&=2^{x-3}\left[{\frac {\log ^{-2+x}(2)}{(-2+x)!}}-\sum _{i=x-1}^{\infty }{\frac {\log ^{i}(2)}{i!}}\right]\\&=2^{x-3}\left[{\frac {\log ^{-2+x}(2)}{(-2+x)!}}-{\frac {2(\Gamma (x-1)-\Gamma (x-1,\log(2)))}{\Gamma (x-1)}}\right]\end{aligned}}$

• $x=3,4,5,\dots$

Recurrence relation

$4\log(2)(\log(4)-x)\Pr(x)+2\left(x^{2}-x(1+\log(2))-2\log ^{2}(2)\right)\Pr(x+1)+x(-x+1+\log(4))\Pr(x+2)=0$

Expected Value

$\mathbb {E} (X)=4$

Variance

$\operatorname {Var} (X)=8\log(4)-10$

Moment Generating Function

$M_{X}(t)={\frac {e^{2t}\left(4^{e^{t}}\left(e^{t}-1\right)+1\right)}{2e^{t}-1}}$

Characteristic Function

$\varphi _{X}(t)={\frac {e^{2it}\left(4^{e^{it}}\left(e^{it}-1\right)+1\right)}{2e^{it}-1}}$

Probability Generating Function

$G(t)={\frac {\left(4^{t}(t-1)+1\right)t^{2}}{2t-1}}$

References

• Aldous, D.J., Fristedt, B., Griffin, P.S., Pruitt, W.E. (1991). The number of extreme points in the convex hull of a random sample. J. of Applied Probability 28, 287-304
• Wimmer, G., Altmann. (1999). Thesaurus of univariate discrete probability distributions. Stamm; 1. ed (1999), pg 7