In probability theory, the law of rare events or Poisson limit theorem states that the Poisson distribution may be used as an approximation to the binomial distribution, under certain conditions. The theorem was named after Siméon Denis Poisson (1781–1840).
As
n
→
∞
and
p
→
0
such that the mean value
n
p
=
λ
remains constant, we can approximate
(
n
k
)
p
k
(
1
−
p
)
n
−
k
≃
e
−
λ
λ
k
k
!
Using Stirling's approximation, we can write:
(
n
k
)
p
k
(
1
−
p
)
n
−
k
=
n
!
(
n
−
k
)
!
k
!
p
k
(
1
−
p
)
n
−
k
≃
2
π
n
(
n
e
)
n
2
π
(
n
−
k
)
(
n
−
k
e
)
n
−
k
k
!
p
k
(
1
−
p
)
n
−
k
=
n
n
−
k
n
n
e
−
k
(
n
−
k
)
n
−
k
k
!
p
k
(
1
−
p
)
n
−
k
Letting
n
→
∞
and
n
p
=
λ
:
(
n
k
)
p
k
(
1
−
p
)
n
−
k
≃
n
n
p
k
(
1
−
p
)
n
−
k
e
−
k
(
n
−
k
)
n
−
k
k
!
=
n
n
(
λ
n
)
k
(
1
−
λ
n
)
n
−
k
e
−
k
n
n
−
k
(
1
−
k
n
)
n
−
k
k
!
=
λ
k
(
1
−
λ
n
)
n
−
k
e
−
k
(
1
−
k
n
)
n
−
k
k
!
≃
λ
k
(
1
−
λ
n
)
n
e
−
k
(
1
−
k
n
)
n
k
!
As
n
→
∞
,
(
1
−
x
n
)
n
→
e
−
x
so:
(
n
k
)
p
k
(
1
−
p
)
n
−
k
≃
λ
k
e
−
λ
e
−
k
e
−
k
k
!
=
λ
k
e
−
λ
k
!
A simpler proof is possible without using Stirling's approximation:
(
n
k
)
p
k
(
1
−
p
)
n
−
k
≃
lim
n
→
∞
n
(
n
−
1
)
(
n
−
2
)
…
(
n
−
k
+
1
)
k
!
(
λ
n
)
k
(
1
−
λ
n
)
n
−
k
=
lim
n
→
∞
n
k
+
O
(
n
k
−
1
)
k
!
λ
k
n
k
(
1
−
λ
n
)
n
−
k
=
lim
n
→
∞
λ
k
k
!
(
1
−
λ
n
)
n
−
k
.
Since
lim
n
→
∞
(
1
−
λ
n
)
n
=
e
−
λ
and
lim
n
→
∞
(
1
−
λ
n
)
−
k
=
1
This leaves
(
n
k
)
p
k
(
1
−
p
)
n
−
k
≃
λ
k
e
−
λ
k
!
.
It is also possible to demonstrate the theorem through the use of Ordinary Generating Functions of the binomial distribution:
G
b
i
n
(
x
;
p
,
N
)
≡
∑
k
=
0
N
[
(
N
k
)
p
k
(
1
−
p
)
N
−
k
]
x
k
=
[
1
+
(
x
−
1
)
p
]
N
by virtue of the Binomial Theorem. Taking the limit
N
→
∞
while keeping the product
p
N
≡
λ
constant, we find
lim
N
→
∞
G
b
i
n
(
x
;
p
,
N
)
=
lim
N
→
∞
[
1
+
λ
(
x
−
1
)
N
]
N
=
e
λ
(
x
−
1
)
=
∑
k
=
0
∞
[
e
−
λ
λ
k
k
!
]
x
k
which is the OGF for the Poisson distribution. (The second equality holds due to the definition of the Exponential function.)
