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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 1 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 1 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