home4_2008

# home4_2008 - POLITECNICO DI TORINO 01KRR STOCHASTIC...

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POLITECNICO DI TORINO 01KRR STOCHASTIC PROCESSES 2008-2009 HOMEWORK 5: DUE TO OCT 13, 2008 Ref . S. Ross, Introduction to Probability models -Eight edition, Chapter 4. Exercises Ross 16 Show that if the state i is recurrent and state i does not communicate with state j , then P ij = 0. This implies that once a process enters a recurrent class of states it can never leave that class. For this reason, a recurrent class is often referred to as a closed class . Ross 20 A transition probability matrix P is said to be doubly stochastic if the sum over each column equals one; that is, X i P ij = 1 , for all j. If such a chain is irreducible and aperiodic and consists of M + 1 states 0 , 1 ,... , show that the limiting probabilities are given by π j = 1 M + 1 , j = 0 , 1 ,...,M. Ross 21 A particle moves on a circle through points which have been marked 0 , 1 , 2 , 3 , 4 (in a clockwise order). At each step it has probability p of moving to the right (clockwise) and 1 - p to the left (counterclockwise). Let

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## This note was uploaded on 10/26/2008 for the course MATH 01KRR taught by Professor Pistone during the Spring '08 term at Politecnico di Torino.

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home4_2008 - POLITECNICO DI TORINO 01KRR STOCHASTIC...

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