8.5 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Displaced Poisson distribution | 1/1 | https://en.wikipedia.org/wiki/Displaced_Poisson_distribution | reference | science, encyclopedia | 2026-05-05T12:22:26.926603+00:00 | kb-cron |
In statistics, the displaced Poisson, also known as the hyper-Poisson distribution, is a generalization of the Poisson distribution.
== Definitions ==
=== Probability mass function === The probability mass function is
P
(
X
=
n
)
=
{
e
−
λ
λ
n
+
r
(
n
+
r
)
!
⋅
1
I
(
r
,
λ
)
,
n
=
0
,
1
,
2
,
…
if
r
≥
0
e
−
λ
λ
n
+
r
(
n
+
r
)
!
⋅
1
I
(
r
+
s
,
λ
)
,
n
=
s
,
s
+
1
,
s
+
2
,
…
otherwise
{\displaystyle P(X=n)={\begin{cases}e^{-\lambda }{\dfrac {\lambda ^{n+r}}{\left(n+r\right)!}}\cdot {\dfrac {1}{I\left(r,\lambda \right)}},\quad n=0,1,2,\ldots &{\text{if }}r\geq 0\\[10pt]e^{-\lambda }{\dfrac {\lambda ^{n+r}}{\left(n+r\right)!}}\cdot {\dfrac {1}{I\left(r+s,\lambda \right)}},\quad n=s,s+1,s+2,\ldots &{\text{otherwise}}\end{cases}}}
where
λ
>
0
{\displaystyle \lambda >0}
and r is a new parameter; the Poisson distribution is recovered at r = 0. Here
I
(
r
,
λ
)
{\displaystyle I\left(r,\lambda \right)}
is the Pearson's incomplete gamma function:
I
(
r
,
λ
)
=
∑
y
=
r
∞
e
−
λ
λ
y
y
!
,
{\displaystyle I(r,\lambda )=\sum _{y=r}^{\infty }{\frac {e^{-\lambda }\lambda ^{y}}{y!}},}
where s is the integral part of r. The motivation given by Staff is that the ratio of successive probabilities in the Poisson distribution (that is
P
(
X
=
n
)
/
P
(
X
=
n
−
1
)
{\displaystyle P(X=n)/P(X=n-1)}
) is given by
λ
/
n
{\displaystyle \lambda /n}
for
n
>
0
{\displaystyle n>0}
and the displaced Poisson generalizes this ratio to
λ
/
(
n
+
r
)
{\displaystyle \lambda /\left(n+r\right)}
.
=== Examples === One of the limitations of the Poisson distribution is that it assumes equidispersion – the mean and variance of the variable are equal. The displaced Poisson distribution may be useful to model underdispersed or overdispersed data, such as:
the distribution of insect populations in crop fields; the number of flowers on plants; motor vehicle crash counts; and word or sentence lengths in writing.
== Properties ==
=== Descriptive Statistics === For a displaced Poisson-distributed random variable, the mean is equal to
λ
−
r
{\displaystyle \lambda -r}
and the variance is equal to
λ
{\displaystyle \lambda }
. The mode of a displaced Poisson-distributed random variable are the integer values bounded by
λ
−
r
−
1
{\displaystyle \lambda -r-1}
and
λ
−
r
{\displaystyle \lambda -r}
when
λ
≥
r
+
1
{\displaystyle \lambda \geq r+1}
. When
λ
<
r
+
1
{\displaystyle \lambda <r+1}
, there is a single mode at
x
=
0
{\displaystyle x=0}
. The first cumulant
κ
1
{\displaystyle \kappa _{1}}
is equal to
λ
−
r
{\displaystyle \lambda -r}
and all subsequent cumulants
κ
n
,
n
≥
2
{\displaystyle \kappa _{n},n\geq 2}
are equal to
λ
{\displaystyle \lambda }
.
== References ==