Biomedical Engineering Reference
In-Depth Information
in Eq. (11.13) is approximated well by Eq. (11.29). Also, the last factor in Eq. (11.13)
can be written
q
N
-
n
=
q
N
(1 -
p
)
N
.
=
(E.15)
Using the binomial expansion for the last expression, we then have
1-
Np
+
N
(
N
-1)
2!
q
N
-
n
p
2
-
=
···
(E.16)
1-
Np
+
(
Np
)
2
2!
=
···=
e
-
Np
.
-
(E.17)
Substitution of Eqs. (11.29) and (E.17) into (11.13) gives
N
n
n
!
p
n
e
-
Np
(
Np
)
n
n
!
e
-
Np
,
(E.18)
P
n
=
=
which is the Poisson distribution, with parameter
Np
.
Normalization
The distribution (E.18) is normalized to unity when summed over
all
non-negative
integers
n
:
∞
e
-
Np
∞
(
Np
)
n
n
!
e
-
Np
e
Np
(E.19)
P
n
=
=
=
1.
n
=
0
n
=
0
For the binomial distribution,
P
n
=
0
when
n
>
N
. As seen from Eq. (E.18), the
terms in the Poisson distribution are never exactly zero.
Mean
Themeanvalueof
n
can be found from Eq. (E.18). With some manipulation of the
summing index
n
,wewrite
µ
≡ e
-
Np
∞
= e
-
Np
∞
n
(
Np
)
n
n
!
n
(
Np
)
n
n
!
(E.20)
n
=
0
n
=
1
e
-
Np
∞
∞
(
Np
)
n
(
n
- 1)!
=
(
Np
)
n
-1
(
n
-1)!
e
-
Np
Np
=
(E.21)
n
=
1
n
=
1
e
-
Np
Np
∞
(
Np
)
n
n
!
e
-
N
p
Np
e
Np
=
=
=
Np
.
(E.22)
n
=
0
The mean of the Poisson distribution is thus identical to that of the binomial dis-
tribution. We write in place of Eq. (E.18) the usual form
P
n
=
µ
n
e
-
µ
n
!
.
(E.23)
