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Aldous distribution Notation
A
l
d
o
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s
(
)
{\displaystyle \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
A
l
d
o
u
s
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x
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≡
Pr
(
X
=
x
)
=
2
x
−
3
[
log
−
2
+
x
(
2
)
(
−
2
+
x
)
!
−
∑
i
=
x
−
1
∞
log
i
(
2
)
i
!
]
=
2
x
−
3
[
log
−
2
+
x
(
2
)
(
−
2
+
x
)
!
−
2
(
Γ
(
x
−
1
)
−
Γ
(
x
−
1
,
log
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2
)
)
)
Γ
(
x
−
1
)
]
{\displaystyle {\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
,
…
{\displaystyle x=3,4,5,\dots }
Recurrence relation
4
log
(
2
)
(
log
(
4
)
−
x
)
Pr
(
x
)
+
2
(
x
2
−
x
(
1
+
log
(
2
)
)
−
2
log
2
(
2
)
)
Pr
(
x
+
1
)
+
x
(
−
x
+
1
+
log
(
4
)
)
Pr
(
x
+
2
)
=
0
{\displaystyle 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
E
(
X
)
=
4
{\displaystyle \mathbb {E} (X)=4}
Variance
Var
(
X
)
=
8
log
(
4
)
−
10
{\displaystyle \operatorname {Var} (X)=8\log(4)-10}
Moment Generating Function
M
X
(
t
)
=
e
2
t
(
4
e
t
(
e
t
−
1
)
+
1
)
2
e
t
−
1
{\displaystyle M_{X}(t)={\frac {e^{2t}\left(4^{e^{t}}\left(e^{t}-1\right)+1\right)}{2e^{t}-1}}}
Characteristic Function
φ
X
(
t
)
=
e
2
i
t
(
4
e
i
t
(
e
i
t
−
1
)
+
1
)
2
e
i
t
−
1
{\displaystyle \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
)
=
(
4
t
(
t
−
1
)
+
1
)
t
2
2
t
−
1
{\displaystyle G(t)={\frac {\left(4^{t}(t-1)+1\right)t^{2}}{2t-1}}}
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
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c498388599903da7508bc17fda34aa50e76d987e)
