Applied probability model homework7
Due day :11/3(Thu) 12:00 pm
1. For the random walk of Example 4.18, use the strong law of large numbers to
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give another proof that the Markov chain is transient when p≠2
2. A transition probability matrix P is said to be doubly stochastic if the sum over
each column equals one ; that is,
P
ij
 1, f o ra l lj.
i
If such a chain is irreducible and aperiodic and consists of M+1 states 0,1,…,M,
show that the limiting probability are given by
j 
1
, j  0,1, . . M
. ,.
M 1
3. A DNA nucleotide has any of four values. A standard model for a mutational
change of the nucleotide at a specific location is a Markov chain model that
supposes that in going from period to period the nucleotide does not change
with probability 1-3α, and if it does change then it is equally likely to change to
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any of the other three values, for some 0<α<3,
(a) Show that P1,n1 
1 3
 (1  4 ) n
4 4
(b) What is the long-run proportion of time the chain is in each state?
I
4. P   0
 R1
0
P1
R2
0
0 , lim P n  ?
n 
Q