In statistics, the displaced Poisson, also known as the hyper-Poisson distribution, is a generalization of the Poisson distribution. 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
where
λ
>
0
and r is a new parameter; the Poisson distribution is recovered at r = 0. Here
I
(
⋅
,
⋅
)
is the incomplete gamma function and 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
)
) is given by
λ
/
n
for
n
>
0
and the displaced Poisson generalizes this ratio to
λ
/
(
n
+
r
)
.
