X and the City: Modeling Aspects of Urban Life (79 page)

Nothing that

we can approximate the logarithm of the factorial
M
! for a large positive integer
M
by a “quick and dirty” method as follows:

From this it follows directly that
M
! ≈
M
^M
e
^−
M
. This result is sometimes known as the weak form of
Stirling’s approximation
. Returning to equation (A4.3) and applying it to the two terms on the right of equation (A4.4), it follows that

Since
n
N
,

Therefore

We have neglected the term in
n
²
since
N
is large. Hence

Therefore, combining the approximations (A4.2) and (A4.5), we have

The expected number of tails in
N
tosses is
λ
=
Np
. Thus equation (A4.6) can be written in terms of the expectation
λ
as

This is the Poisson distribution for the probability of exactly
n
tails occurring in
N
tosses of the coin, when
p
1. And instead of heads and tails, we can think of customers arriving at the post office, or gaps in traffic, as applied in
Chapters 3
and
9
, respectively.

We can think of this in another way. Note that

This result can be interpreted as the sums of the probabilities of all possible outcomes occurring; in other words, we have a probability distribution.

We state without proof a rather formal expression of a general problem to which this method applies:

Any local maxima or minima of a function
z
=
f
(
x
,
y
) that is subject to the constraint
g
(
x
,
y
) = 0 will be among those points (
x
₀
,
y
₀
) for which the point (
x
₀
,
y
₀
,
λ
₀
) is a solution to the system of equations