594
A Aspects of Combinatorial Analysis
This result is often written in the form
p(n) =
N!
n!(N − n)!
p
n q
N −n ,
(A.1.10)
and is known as the binomial distribution. This name is based upon the well-known
binomial theorem
(x + y)
n
=
n
m=0
n
m
x
m y
n−m .
(A.1.11)
The Gaussian distribution
p(N, n) = p(N, n) e
−(n−n) 2 /(2Npq) ,
N
0
dn p(N, n) = 1 ,
(A.1.12)
is the continuous variable equivalent of the binomial distribution. The value of
p(N, n) is obtained from the normalization condition, which gives
p(N, n)
N
0
dn e
−(n−n) 2 /(Npq)
= 1 .
Evaluation of this integral proceeds by changing variables from n to y = n − n, so
that the integral over n from 0 to N is replaced by an integral over y from −n to
N − n. For N sufficiently large (say, of order 10 6 or more), it can be replaced by
N −n
−n
dy e
−y 2 /(2Npq)
∞
−∞
dy e
−y 2 /(2Npq)
= 2
∞
0
dy e
−y 2 /(2Npq) ,
with the latter step arising from the even symmetry under inversion (i.e., y → −y)
on the interval [−a, a] over which the integration is being performed. The definite
integral appearing in the final expression has the form
∞
0 e −ax 2 dx, which can be
obtained from a table of definite integrals. 1 Similar arguments may be employed for
the evaluation of the mean value, n, of n and the variance, (n − n) 2 , for n.
1 It may also be evaluated directly by noting that its square can be written as the product of the same
integral over the interval [0,∞) for x and for y, i.e., as the double integral over the positive quadrant
of the xy-plane of the integrand e a(x 2 +y 2 ) : this double integral can then be converted into polar
coordinates and readily evaluated as π/(4a), whose square root gives
∞
0 e −ax 2 dx =
1
2
√
π/a.
A Aspects of Combinatorial Analysis
This result is often written in the form
p(n) =
N!
n!(N − n)!
p
n q
N −n ,
(A.1.10)
and is known as the binomial distribution. This name is based upon the well-known
binomial theorem
(x + y)
n
=
n
m=0
n
m
x
m y
n−m .
(A.1.11)
The Gaussian distribution
p(N, n) = p(N, n) e
−(n−n) 2 /(2Npq) ,
N
0
dn p(N, n) = 1 ,
(A.1.12)
is the continuous variable equivalent of the binomial distribution. The value of
p(N, n) is obtained from the normalization condition, which gives
p(N, n)
N
0
dn e
−(n−n) 2 /(Npq)
= 1 .
Evaluation of this integral proceeds by changing variables from n to y = n − n, so
that the integral over n from 0 to N is replaced by an integral over y from −n to
N − n. For N sufficiently large (say, of order 10 6 or more), it can be replaced by
N −n
−n
dy e
−y 2 /(2Npq)
∞
−∞
dy e
−y 2 /(2Npq)
= 2
∞
0
dy e
−y 2 /(2Npq) ,
with the latter step arising from the even symmetry under inversion (i.e., y → −y)
on the interval [−a, a] over which the integration is being performed. The definite
integral appearing in the final expression has the form
∞
0 e −ax 2 dx, which can be
obtained from a table of definite integrals. 1 Similar arguments may be employed for
the evaluation of the mean value, n, of n and the variance, (n − n) 2 , for n.
1 It may also be evaluated directly by noting that its square can be written as the product of the same
integral over the interval [0,∞) for x and for y, i.e., as the double integral over the positive quadrant
of the xy-plane of the integrand e a(x 2 +y 2 ) : this double integral can then be converted into polar
coordinates and readily evaluated as π/(4a), whose square root gives
∞
0 e −ax 2 dx =
1
2
√
π/a.