accomplished by expressing P2(i I n) as a sum over all j of the corresponding P,(j n)
times a factor which accounts for the probability of the collection of mutation events
required to transform j into i. This probability can be expressed as p(i,)(1 pm)L-H(iJ)
where H(i,j) = H(j, i) is the Hamming distance of the pair i,j. That is, H is a function
defined on S x S with values in {0, 1,2, --,L}. H(i,j) is the number of bits which must be
altered by mutation to transform i into j and L H(i,j) is the number of bits which must
remain unaltered. Thus, P2(i I n) can be written as
Vi e S, Vn e S' P2(i I n) = p)() pm)L-(i) PI(j I n)
jes
mr I(ij)
=(1-pm)Lj (1 xP,(j|n) Eq.4.11
= 1 satIj) x Pi l n)
(1+ a)Lj e
where
Pm
a=- Eq. 4.12
(1 -Pm)
and
Pm- ( Eq. 4.13
(1 + a)
For pm=0 or pm=l, Eq. 4.11 includes the indeterminate form 00 in some terms. Thus,
the admissible range of pm is restricted to 0 < p, < 1, and consequently that of a is
0 < a < oo. However, cases corresponding to pm > 1/2 = a> > 1 are of no practical interest
(they are less random than the case pm = 1/2 = a = 1), and some of the following devel-
opments restrict consideration to the range 0 < pm 1/2 :* 0 < a < 1.
Substituting Eq. 4.2 into Eq. 4.11 yields
1 n((iJ) nO) x RO)
P2(i I n)= a x j) (
(1 + a)L s E n(k) x R(k)
k S
It is also straightforward to show that