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Abakuks Notation
A
b
a
k
u
k
s
(
a
,
r
,
n
)
{\displaystyle \mathrm {Abakuks} (a,r,n)}
Parameters
a >0, r ∈ { 2, 3, ... , n} , n ∈ { 2, 3, 4, ... } Support
x ∈ { r-1, r, r+1, ... , 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 }
probability mass function of the Abakuks distribution is
A
b
a
k
u
k
s
(
x
;
a
,
r
,
n
)
≡
Pr
(
X
=
x
)
=
1
x
(
a
n
)
x
(
n
−
x
)
!
T
{\displaystyle \mathrm {Abakuks} (x;a,r,n)\equiv \Pr(X=x)={\frac {1}{x(an)^{x}(n-x)!T}}}
x
=
r
−
1
,
r
,
r
+
1
,
.
.
.
,
n
{\displaystyle x=r-1,r,r+1,...,n}
n
∈
N
+
−
{
1
}
{\displaystyle n\in \mathbb {N} ^{+}-\{1\}}
r
∈
{
2
,
3
,
,
.
.
.
,
n
}
{\displaystyle r\in \{2,3,,...,n\}}
a
>
0
{\displaystyle a>0}
T
=
∑
j
=
r
−
1
n
1
j
(
a
n
)
j
(
n
−
j
)
!
{\displaystyle T=\sum _{j\mathop {=} r-1}^{n}{\frac {1}{j(an)^{j}(n-j)!}}}
Recurrence relation
{
a
n
(
x
+
1
)
Pr
(
x
+
1
)
+
(
x
2
−
n
x
)
Pr
(
x
)
=
0
}
{\displaystyle \left\{an(x+1)\Pr(x+1)+\left(x^{2}-nx\right)\Pr(x)=0\right\}}
Expected Value
E
[
X
]
=
a
n
(
1
a
n
)
r
2
F
0
(
1
,
−
n
+
r
−
1
;
;
−
1
a
n
)
T
Γ
(
n
−
r
+
2
)
{\displaystyle \operatorname {E} [X]={\frac {an\left({\frac {1}{an}}\right)^{r}\,_{2}F_{0}\left(1,-n+r-1;;-{\frac {1}{an}}\right)}{T\Gamma (n-r+2)}}}
Moment Generating Function
M
X
(
t
)
=
Γ
(
r
−
1
)
(
e
t
a
n
)
r
−
1
3
F
~
1
(
1
,
r
−
1
,
−
n
+
r
−
1
;
r
;
−
e
t
a
n
)
T
Γ
(
n
−
r
+
2
)
{\displaystyle M_{X}(t)={\frac {\Gamma (r-1)\left({\frac {e^{t}}{an}}\right)^{r-1}\,_{3}{\tilde {F}}_{1}\left(1,r-1,-n+r-1;r;-{\frac {e^{t}}{an}}\right)}{T\Gamma (n-r+2)}}}
Characteristic Function
φ
X
(
t
)
=
Γ
(
r
−
1
)
(
e
i
t
a
n
)
r
−
1
3
F
~
1
(
1
,
r
−
1
,
−
n
+
r
−
1
;
r
;
−
e
i
t
a
n
)
T
Γ
(
n
−
r
+
2
)
{\displaystyle \varphi _{X}(t)={\frac {\Gamma (r-1)\left({\frac {e^{it}}{an}}\right)^{r-1}\,_{3}{\tilde {F}}_{1}\left(1,r-1,-n+r-1;r;-{\frac {e^{it}}{an}}\right)}{T\Gamma (n-r+2)}}}
Probability Generating Function
G
(
t
)
=
(
t
a
n
)
r
−
1
3
F
1
(
1
,
r
−
1
,
−
n
+
r
−
1
;
r
;
−
t
a
n
)
(
r
−
1
)
T
(
n
−
r
+
1
)
!
{\displaystyle G(t)={\frac {\left({\frac {t}{an}}\right)^{r-1}\,_{3}F_{1}\left(1,r-1,-n+r-1;r;-{\frac {t}{an}}\right)}{(r-1)T(n-r+1)!}}}
Interrelations [ edit ]
Symbol
Meaning
∼
{\displaystyle \sim }
X
∼
Y
{\displaystyle X\sim Y}
≡
{\displaystyle \equiv }
the distribution in the title is identical with this distribution
⇐
{\displaystyle \Leftarrow }
the distribution in title is a special case of this distribution
⇒
{\displaystyle \Rightarrow }
this distribution is a special case of the distribution in the title
←
{\displaystyle \leftarrow }
this distribution converges to the distribution in the title
→
{\displaystyle \rightarrow }
the distribution in the title converges to this distribution
Relationship
Distribution
When
⇐
{\displaystyle \Leftarrow }
generalized power series family
(
a
x
,
θ
,
f
(
.
)
,
T
′
)
{\displaystyle \left(a_{x},\theta ,f(.),T'\right)}
a
x
=
1
x
(
n
)
x
(
n
−
x
)
!
θ
=
1
a
f
(
θ
)
=
∑
j
=
r
−
1
n
1
j
(
a
n
)
j
(
n
−
j
)
!
T
T
′
=
{
r
−
1
,
r
,
.
.
.
,
n
}
{\displaystyle a_{x}={\frac {1}{x(n)^{x}(n-x)!}}\qquad \theta ={\frac {1}{a}}\qquad f(\theta )=\sum _{j\mathop {=} r-1}^{n}{\frac {1}{j(an)^{j}(n-j)!T}}\qquad T'=\{r-1,r,...,n\}}
⇐
{\displaystyle \Leftarrow }
(
r
−
1
)
{\displaystyle (r-1)}
(
a
i
,
b
j
,
θ
,
k
,
r
)
{\displaystyle \left(a_{i},b_{j},\theta ,k,r\right)}
k
=
2
r
=
1
θ
=
−
1
a
n
{\displaystyle k=2\qquad r=1\qquad \theta ={\frac {-1}{an}}}
⇐
{\displaystyle \Leftarrow }
Kemp-Dacey-hypergeometric family (II/20)
(
I
I
/
20
)
{\displaystyle (II/20)}
References [ edit ] Ababuks, A. (1976). An invariance property of the Poisson distributions for a limited immigrant population. J. of the Royal Statistical Society B 38, 99-101 Wimmer, G., Altmann. (1999). Thesaurus of univariate discrete probability distributions. Stamm; 1. ed (1999) , pg 1
![{\displaystyle \operatorname {E} [X]={\frac {an\left({\frac {1}{an}}\right)^{r}\,_{2}F_{0}\left(1,-n+r-1;;-{\frac {1}{an}}\right)}{T\Gamma (n-r+2)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/363113ac6939d056bc49f467e1e5c1b442abf63e)
