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Ahmad-Kudo-Poisson distribution Notation
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{\displaystyle \mathrm {AhmadKudoPoisson} (a,c,\alpha )}
Support
x ∈ { 0, 1, 2, ... } 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 modified Poisson distribution , the probability mass function of the Ahmad-Kudo-Poisson distribution is given by
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Pr
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{\displaystyle \mathrm {AhmadKudoPoisson} (x;a,c,\alpha )\equiv \Pr(X=x)={\begin{cases}{\frac {e^{-a}a^{x}}{x!}}&x=0,1,\dots ,c-1,\\{\frac {e^{-a}(1-\alpha )a^{c}}{c!}}&x=c,\\{\frac {e^{-a}a^{c+1}\left({\frac {\alpha (c+1)}{a}}+1\right)}{(c+1)!}}&x=c+1,\\{\frac {e^{-a}a^{x}}{x!}}&x=c+2,c+3,\dots \\\end{cases}}}
x
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0
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1
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2
,
…
{\displaystyle x=0,1,2,\dots }
a
≥
0
{\displaystyle a\geq 0}
c
∈
N
{\displaystyle c\in \mathbb {N} }
0
≤
α
≤
1
{\displaystyle 0\leq \alpha \leq 1}
Cumulative Distribution Function
F
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=
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Γ
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Γ
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Γ
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Γ
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x
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Γ
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Γ
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Γ
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{\displaystyle F_{X}(x)={\begin{cases}{\frac {\Gamma (x+1,a)}{\Gamma (x+1)}}&x=0,1,\dots ,c-1,\\{\frac {e^{-a}(1-\alpha )a^{c}}{c!}}+{\frac {\Gamma (c,a)}{\Gamma (c)}}&x=c,\\{\frac {e^{-a}a^{c+1}\left({\frac {\alpha (c+1)}{a}}+1\right)}{(c+1)!}}+{\frac {e^{-a}(1-\alpha )a^{c}}{c!}}+{\frac {\Gamma (c,a)}{\Gamma (c)}}&x=c+1,\\{\frac {\Gamma (x+1,a)}{\Gamma (x+1)}}&x=c+2,c+3,\dots \\\end{cases}}}
Expected Value
E
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a
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Γ
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c
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1
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{\displaystyle \mathbb {E} (X)=a+{\frac {e^{-a}\alpha a^{c}}{\Gamma (c+1)}}}
Variance
Var
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)
=
a
+
e
−
2
a
α
a
c
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e
a
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−
2
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Γ
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Γ
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2
{\displaystyle \operatorname {Var} (X)=a+{\frac {e^{-2a}\alpha a^{c}\left(e^{a}(-2a+2c+1)\Gamma (c+1)-\alpha a^{c}\right)}{\Gamma (c+1)^{2}}}}
Moment Generating Function
M
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=
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{\displaystyle M_{X}(t)={\frac {e^{-a}\alpha \left(e^{t}-1\right)\left(ae^{t}\right)^{c}}{c!}}+e^{a\left(e^{t}-1\right)}}
Characteristic Function
φ
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{\displaystyle \varphi _{X}(t)={\frac {e^{-a}\alpha \left(e^{it}-1\right)\left(ae^{it}\right)^{c}}{c!}}+e^{a\left(e^{it}-1\right)}}
Probability Generating Function
G
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=
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{\displaystyle G(t)={\frac {e^{-a}\alpha (t-1)(at)^{c}}{c!}}+e^{a(t-1)}}
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 \Rightarrow }
deterministic
a
=
0
{\displaystyle a=0}
⇒
{\displaystyle \Rightarrow }
Poisson
(
a
)
{\displaystyle \left(a\right)}
α
=
0
{\displaystyle \alpha =0}
Ahmad, M., Kudo, A. (1967). Modified and partially truncated Poisson distribution Bulletin of the Institute of Statistical Research and Training, University of Dacca 1, 82-90 Wimmer, G., Altmann. (1999). Thesaurus of univariate discrete probability distributions. Stamm; 1. ed (1999) , pg 5
