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Markov process and irreducible matrix

**Question 1**:

a) Describe about Markov process, ergodic chain and irreducible matrix?

b) which of the given matrices are stochastic?

**Question 2**:

a) state and prove changing stakes result x_{2}

b) If p = 1/3, q = 1/2, z =1, a = 1000 prove that d_{z} = 999.

**Question 3**:

a) A fair coin is tossed repeatedly. If X_{n} denote the maximum number of numbers occurring in the first n tosses, find the transition probability matrx P of the Markov chain. Also find P^{2} and P (X_{2} = 6)

b) which of the given matrices are regular.

**Question 4**: Three boys A, B and C are throwing a ball to each other. A always throws the ball to B and B always throws to C; but C is just as likely to throw the ball to B as to A. Show that the process is Markovian. Find out the transition matrix and categorize the states.

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