Question

Question 2: Discrete time Markov chains jo eo L a nosno og Supaduioo are 'X aq ua o1) sorueduso0 OML x Auedusoo qa sds Auedu0o sagaad Janosno V reas Aaa sreop uo9 OIS year with probability 80 % . A customer that prefers company Y stays with company Y next year with probability 75 %. (b) Assume a customer is using company X. What is the probability they A Auedumoo q ae (c) What is the probability that a random customer is with company X and what is the prob- ability that a random customer is with company Y?

Answer 1

(a) The discrete time series can be modelled as below. The probabilities in the matrix represent the hance of transition from current the state to the futre state. Future State X Current X 0.8 0.2 equal to T (Say) State Y 0.25 0.75 The TPM after two time periods can be found by finding calculating the squate of the matrix. So T is calculated to be: (b) Y 0.31 T Square X 0.69 0.6125 0.3875 Y The probailty that a customer from X moves to Y in two months is 0.31. Let (C) The probability that the customer is with company X The probability that the customer is with company Y P1 P2 P1 and P2 are to be estimated in such a way that (P1, P2) * T = (P1, P2) 0.2 is equal to 0.8 0.75 (P1, P2) 0.25 (P1, P2) i.e. 0.8P1 + 0.25P2 P1 and 0.2P1 + 0.75P2 = P2 P1+P2 = 1. Substituting P2 = 1-P1, we find, 0.45 P1 0.25, which implies P1 = 0.56 and P1 = 0.44