 | ||
Parameters λ
>
0
{\displaystyle \lambda >0}
(fixed mean)
α
,
β
>
0
{\displaystyle \alpha ,\beta >0}
(parameters of variable mean) Support k
∈
{
0
,
1
,
2
,
…
}
{\displaystyle k\in \{0,1,2,\ldots \}} pmf ∑
i
=
0
k
Γ
(
α
+
i
)
β
i
λ
k
−
i
e
−
λ
Γ
(
α
)
i
!
(
1
+
β
)
α
+
i
(
k
−
i
)
!
{\displaystyle \sum _{i=0}^{k}{\frac {\Gamma (\alpha +i)\beta ^{i}\lambda ^{k-i}e^{-\lambda }}{\Gamma (\alpha )i!(1+\beta )^{\alpha +i}(k-i)!}}} CDF ∑
j
=
0
k
∑
i
=
0
j
Γ
(
α
+
i
)
β
i
λ
j
−
i
e
−
λ
Γ
(
α
)
i
!
(
1
+
β
)
α
+
i
(
j
−
i
)
!
{\displaystyle \sum _{j=0}^{k}\sum _{i=0}^{j}{\frac {\Gamma (\alpha +i)\beta ^{i}\lambda ^{j-i}e^{-\lambda }}{\Gamma (\alpha )i!(1+\beta )^{\alpha +i}(j-i)!}}} Mean λ
+
α
β
{\displaystyle \lambda +\alpha \beta } Mode {
z
,
z
+
1
{
z
∈
Z
}
:
z
=
(
α
−
1
)
β
+
λ
⌊
z
⌋
otherwise
{\displaystyle {\begin{cases}z,z+1&\{z\in \mathbb {Z} \}:\;z=(\alpha -1)\beta +\lambda \\\lfloor z\rfloor &{\textrm {otherwise}}\end{cases}}} | ||
The Delaporte distribution is a discrete probability distribution that has received attention in actuarial science. It can be defined using the convolution of a negative binomial distribution with a Poisson distribution. Just as the negative binomial distribution can be viewed as a Poisson distribution where the mean parameter is itself a random variable with a gamma distribution, the Delaporte distribution can be viewed as a compound distribution based on a Poisson distribution, where there are two components to the mean parameter: a fixed component, which has the
Properties
The skewness of the Delaporte distribution is:
The excess kurtosis of the distribution is: