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4.1.2
Poisson webs
The binary nature of the above process naturally leads to the binomial distribution, even
if it has a somewhat different form from what we are accustomed to seeing. Consider
the asymmetric binary process where
p
is the probability of realizing outcome 1 and
1
p
is the probability of realizing outcome 2. In this case the probability of realizing
the first outcome
m
times out of
N
trials can be obtained from the binomial expansion
−
N
N
(
1
−
p
+
p
)
=
P
N
(
m
),
(4.10)
m
=
0
where the probability is determined by the binomial expansion to be
N
!
p
m
N
−
m
P
N
(
m
)
=
(
1
−
p
)
.
(4.11)
(
N
−
m
)
!
m
!
When the number of events is very large and the probability of a particular outcome
is very small the explicit form (
4.11
) is awkward to deal with and approximations to
the exact expression are more useful. This was the case in the last section when the
probability of each outcome was the same for very large
N
and the Gaussian distribution
was the approximate representation of the binomial distribution.
Suppose that we have
N
disagreements among various groups in the world, such
as between nations, unions and management, etc. Suppose further that the probability
of any one of these disagreements undergoing a transition into a “deadly quarrel” in a
given interval of time is
p
. What is the probability that
m
of the disagreements undergo
transitions to deadly quarrels in the time interval? The Quaker scientist Richardson
[
39
] determined that the distribution describing the number of “deadly quarrels” is the
Poisson distribution, so that from the conditions of his argument we can extract the
Poisson distribution as another approximation to the above binomial distribution.
The first factor on the rhs of (
4.11
) has the explicit form
N
!
)
!
=
N
(
N
−
1
)...(
N
−
m
+
1
),
(4.12)
(
N
−
m
which has
m
factors, each of which differs from
N
by terms of order
m
/
N
1 and can
therefore be neglected, yielding
N
!
N
m
)
!
≈
.
(4.13)
(
N
−
m
Replacing the ratio of factorials in the binomial distribution with this expression yields
N
m
m
p
m
N
−
m
P
N
(
m
)
≈
(
1
−
p
)
(4.14)
!
and in the limit of very large
N
and very small
p
, but such that their product is constant,
λ
=
,
Np
(4.15)
the approximate binomial (
4.14
) becomes
m
)
≈
λ
N
−
m
P
N
(
m
!
(
1
−
p
)
.
(4.16)
m
