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Unformatted text preview: he states x = 0 and N that correspond to ﬁxation of the
population in the all a or all A states are absorbing states, that is, p(x, x) = 1.
So it is natural to ask:
Q1. Starting from i of the A alleles and N i of the a alleles, what is the
probability that the population ﬁxates in the all A state?
To make this simple model more realistic we can introduce the possibility
of mutations: an A that is drawn ends up being an a in the next generation
with probability u, while an a that is drawn ends up being an A in the next
generation with probability v . In this case the probability an A is produced by
a given draw is
i
Ni
⇢i = (1 u) +
v
N
N
but the transition probability still has the binomial form
✓◆
N
p(i, j ) =
(⇢i )j (1 ⇢i )N j
j
If u and v are both positive, then 0 and N are no longer absorbing states,
so we ask:
Q2. Does the genetic composition settle down to an equilibrium distribution as
time t ! 1? As the next example shows it is easy to extend the notion of a Markov chain
to cover situations in which the future evolution is independent of the past when
we know the last two states.
Example 1.10. Twostage M...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).
 Spring '10
 DURRETT
 The Land

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