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The probability mass function of the Barlett distribution is given by
B
a
r
l
e
t
t
(
x
;
a
,
q
)
≡
Pr
(
X
=
x
)
=
∑
j
=
0
x
a
−
j
+
x
e
−
a
q
j
p
(
−
j
+
x
)
!
=
p
e
a
p
q
q
x
Γ
(
x
+
1
,
a
q
)
x
!
{\displaystyle {\begin{aligned}\mathrm {Barlett} (x;a,q)\equiv \Pr(X=x)&=\sum _{j=0}^{x}{\frac {a^{-j+x}e^{-a}q^{j}p}{(-j+x)!}}\\&={\frac {pe^{\frac {ap}{q}}q^{x}\Gamma \left(x+1,{\frac {a}{q}}\right)}{x!}}\end{aligned}}}
x
=
0
,
1
,
2
,
…
{\displaystyle x=0,1,2,\dots }
a
≥
0
{\displaystyle a\geq 0}
0
<
p
≤
1
{\displaystyle 0<p\leq 1}
q
=
1
−
p
{\displaystyle q=1-p}
Expected Value
E
[
X
]
=
a
+
1
p
−
1
{\displaystyle \operatorname {E} [X]=a+{\frac {1}{p}}-1}
Variance
Var
(
X
)
=
a
+
q
p
2
{\displaystyle \operatorname {Var} (X)=a+{\frac {q}{p^{2}}}}
Moment Generating Function
M
X
(
t
)
=
p
e
a
(
e
t
−
1
)
1
−
q
e
t
{\displaystyle M_{X}(t)={\frac {pe^{a\left(e^{t}-1\right)}}{1-qe^{t}}}}
Characteristic Function
φ
X
(
t
)
=
p
e
a
(
e
i
t
−
1
)
1
−
q
e
i
t
{\displaystyle \varphi _{X}(t)={\frac {pe^{a\left(e^{it}-1\right)}}{1-qe^{it}}}}
Probability Generating Function
G
(
t
)
=
p
e
a
(
t
−
1
)
1
−
q
t
{\displaystyle G(t)={\frac {pe^{a(t-1)}}{1-qt}}}
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 \equiv }
Poisson
(
a
)
∗
{\displaystyle (a)*}
(
q
)
{\displaystyle (q)}
≡
{\displaystyle \equiv }
Poisson
(
a
−
log
(
p
)
)
∨
{\displaystyle (a-\log(p))\vee }
(
a
′
,
H
(
.
)
)
{\displaystyle (a',H(.))}
a
′
=
a
−
log
(
p
)
H
(
t
)
=
a
t
−
log
(
1
−
q
t
)
a
−
log
(
p
)
{\displaystyle a'=a-\log(p)\qquad H(t)={\frac {at-\log(1-qt)}{a-\log(p)}}}
⇐
{\displaystyle \Leftarrow }
Charlier
(
a
′
,
b
,
m
,
β
)
{\displaystyle (a',b,m,\beta )}
a
′
=
m
=
1
b
=
a
β
=
q
{\displaystyle a'=m=1\qquad b=a\qquad \beta =q}
⇐
{\displaystyle \Leftarrow }
Lüders
(
a
′
,
b
,
c
)
{\displaystyle (a',b,c)}
a
′
−
b
c
=
a
c
=
q
b
=
q
2
{\displaystyle a'-{\frac {b}{c}}=a\qquad c=q\qquad b=q^{2}}
⇐
{\displaystyle \Leftarrow }
Poisson-negative binomial convolution
(
a
,
k
,
p
)
{\displaystyle (a,k,p)}
k
=
1
{\displaystyle k=1}
←
{\displaystyle \leftarrow }
multiple Poisson
(
n
,
a
i
)
{\displaystyle (n,a_{i})}
a
1
=
a
+
q
a
i
=
q
i
i
n
→
∞
{\displaystyle a_{1}=a+q\qquad a_{i}={\frac {q^{i}}{i}}\qquad n\rightarrow \infty }
⇒
{\displaystyle \Rightarrow }
geometric
(
q
)
{\displaystyle (q)}
a
=
0
{\displaystyle a=0}
⇒
{\displaystyle \Rightarrow }
Poisson
(
q
)
{\displaystyle (q)}
p
=
1
{\displaystyle p=1}
Bartlett, M.S. (1969). Distributions associated with cell populations. Biometrika 56, 391-400. Berg, S., Jaworski, J. (1988). Modified binomial and Poisson distributions with applications in random mapping theory. J. of Statistical Planning and Inference 18, 313-322. Samaniego, F.J. (1976). A characterization of convoluted Poisson distributions with applications to estimation. J. of the American Statistical Association 71, 475-479. Wimmer, G., Altmann. (1996a). The multiple Poisson distribution, its characteristics and a variety of forms. Biometrical J. 8, 995-1011 Wimmer, G., Altmann. (1999). Thesaurus of univariate discrete probability distributions. Stamm; 1. ed (1999) , pg 14 
![{\displaystyle \operatorname {E} [X]=a+{\frac {1}{p}}-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95be7758da52c0e856cae786c439a264b517a2e7)
