Denker
Spring 2010
416 Stochastic Modeling  Assignment 6
Solution
Problem 1:
(Problem 44, p. 271) Suppose that a population consists of a fixed number
of genes, say
m
genes. If any generation has exactly
i
of its
m
genes of type 1, then the
next generation will have
j
type 1 genes (and hence
m
−
j
type 2 genes) with probability
parenleftbigg
m
j
parenrightbigg
i
j
(
m
−
i
)
m

j
m

m
.
Let
X
n
denote the number of type 1 genes in the nth generation, and assume that
X
0
=
i
.
1. Find
E
[
X
n
].
2. What is the probability that eventually all the genes will be type 1?
Solution:
We have that
P
(
X
n
+1
=
j

X
n
=
i
) =
parenleftbigg
m
j
parenrightbigg parenleftbigg
i
m
parenrightbigg
j
parenleftbigg
m
−
i
m
parenrightbigg
m

j
.
1. Hence
E
[
x
n
+1

X
n
=
i
] =
m
i
m
=
i
=
X
n
,
and
E
[
X
n
+1
] =
summationdisplay
i
E
[
X
n
+1

X
n
=
i
]
P
(
X
n
=
i
) =
E
[
X
n
]
.
Therefore
E
[
X
n
] =
E
[
X
0
] =
i
, since the chain starts in
X
0
= 1 with probability one.
2. 0 and
m
are absorbing states, all other states are transient. Hence lim
X
n
= 0 or =
m
.
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 Spring '08
 DENKER
 Probability, Stochastic Modeling, Probability theory, Stochastic process, yn, Markov chain, Xn

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